' 


ESSENTIALS    OF 

MECHANICAL   DRAFTING 

ELEMENTS,   PRINCIPLES,    AND    METHODS 

WITH    SPECIFIC    APPLICATIONS    IN    WORKING    DRAWINGS    OF 
FURNITURE,    MACHINE,    AND    SHEET    METAL    CONSTRUCTION 

A   MANUAL  FOR   STUDENTS 

ARRANGED    FOR    REFERENCE    AND     STUDY    IN    CONNEC- 
TION WITH  COURSES  IN  MANUAL  TRAINING,  INDUSTRIAL, 
HIGH,  AND  TECHNICAL  SCHOOLS 


BY 

LUDWIG   FRANK 

INSTRUCTOR  IN  DRAWING,  HIGH  SCHOOL  OF  COMMERCE,  BOSTON,  MASS. 
FORMERLY  ASSISTANT  TO  THE  DIRECTOR  OF  MANUAL  ARTS,  BOSTON,  MASS. 
SUPERVISING  DRAWING  IN  PREVOCATIONAL,  INDUSTRIAL,  AND  HIGH  SCHOOLS 


1917 
MILTON  BRADLEY  COMPANY 

SPRINGFIELD,    MASSACHUSETTS 


COPYRIGHT  1917 

BY 

MILTON  BRADLEY  COMPANY, 
SPRINGFIELD,  MASS. 


PREFACE 

The  purpose  of  this  book  is  to  provide  the  student  with  definite  comprehensive 
text  and  illustrations  comprising  the  theory  and  practice  of  mechanical  drafting, 
which  shall  effectively  supplement  and  give  emphasis  to  the  work  of  the  teacher, 
and  at  the  same  time  afford  complete  freedom  in  the  presentation  and  application 
of  principles  to  meet  different  requirements,  conditions,  and  individual  needs. 

In  view  of  this  purpose,  and  for  greater  convenience  of  reference  and  connected 
study  of  related  subject-matter,  the  text  is  presented  in  a  progressive  series  of 
topically  arranged  articles  with  appropriate  cross  references. 

This  arrangement  is  not  intended,  however,  as  a  prescribed  order  of  study 
to  be  rigidly  adhered  to,  nor  is  the  content  of  the  text  intended  to  supersede 
necessary  personal  instruction  or  thoughtful  study  on  the  part  of  the  student. 

The  book  is  designed  to  be  used  as  the  teacher  may  determine  in  connection 
with  his  own  course;  to  conserve  the  time  and  energy  usually  expended  in  repeti- 
tion; to  secure  a  systematic  study  of  such  text  and  illustrations  as  relate  to  the 
oral  presentation;  and  to  enable  the  student  to  review  any  desired  topic  as 
individual  need  arises  and  to  proceed  with  the  minimum  of  dependence  upon 
the  teacher. 

It  is  believed  that  the  content  and  arrangement  will  be  found  adaptable  and 
adequate  wherever  mechanical  drafting  is  taught;  will  assist  the  teacher  in  formu- 
lating specific  courses;  will  stimulate  the  interest  of  the  student  by  giving  a 
greater  appreciation  of  the  utility  and  scope  of  the  subject;  and  will  prove  an 
efficient  aid  in  developing  a  working  knowledge  of  the  elements,  principles,  and 
methods  of  drafting  as  applied  in  general  practice. 

The  writer  gratefully  acknowledges  his  obligations  to  numerous  engineers  and 
draftsmen,  and  to  the  following  directors  and  teachers  of  Boston,  for  many  helpful 
suggestions:  Mr.  Arthur  L.  Williston,  Director  of  Wentworth  Institute; 
Mr.  John  C.  Brodhead,  Associate  Director  of  Manual  Arts;  Mr.  Edw.  C.  Emerson, 
Assistant  Director  of  Manual  Arts;  Messrs.  R.  H.  Knapp  and  E.  H.  Temple 
of  the  Mechanic  Arts  High  School;  Messrs.  A.  Roswall  and  E.  M.  Longfield  of  the 
Boys'  Industrial  School;  Mr.  A.  L.  Primeau,  formerly  of  the  South  Boston  Pre- 
vocational  Center;  and  Mr.  R.  A.  Day  of  the  Hyde  Park  Co-operative  Classes. 
Messrs.  Temple,  Primeau,  and  Day  also  gave  valuable  assistance  in  the  prepara- 
tion of  the  drawings. 

LUDWIG  FRANK. 
Brookline,  January,  1917. 


2065942 


CONTENTS 


CHAPTER  I 
INTRODUCTION 


ART.  PAGE 

1.  Nature  and  Uses  of  Mechanical  Draft- 

ing        1 

2.  Geometric  Terms,  Definitions,  etc. ...       2 


3.  General  Instructions  for  Working  Out 

Problems 11 


CHAPTER  II 
INSTRUMENTS,  MATERIALS,  AND  THEIR  USE. 


4.  List  of  Equipment 13 

5.  Care  and  Arrangement  of  Equipment  13 

6.  Drawing  Board 14 

7.  T-Square 14 

8.  Triangles 15 

9.  Pencils  and  Writing  Pens 16 

10.  Needle-point 17 

11.  Scales 17 


12.  Protractor 

13.  Compasses 

14.  Bow  Compasses. . . . 

15.  Dividers 

16.  Bow  Dividers. 

17.  Ruling  Pen 

18.  Curve  Rulers 

19.  Erasers . .  , 


19 
19 
20 
20 
20 
20 
21 
22 


CHAPTER  III 
PENCILING  AND  FINISH  RENDERING 


20.  Layout  of  the  Sheet 23 

21.  Constructive  Stage  of  the  Drawing. . .     23 

22.  Finishing  Stage  of  the  Drawing 24 


23.  Shadow  Lining 25 

24.  Line  Shading 28 

25.  Lettering 29 


CHAPTER  IV 
GEOMETRIC   CONSTRUCTION 


26.  Geometric  and  Practical  Methods 33 

27.  To  bisect  a  straight  line  or  a  circular  arc    33 

28.  To  bisect  an  angle 33 

29.  To  construct  an  angle  equal  to  a  given 

angle 34 

30.  To   divide   a   straight   line   into  any 

number  of  equal  parts 34 

31.  To  divide  a  straight  line  into  parts 

proportional  to  those  of  a  given 
divided  line 34 

32.  To  layoff  the  length  of  a  given  circular 

curve  upon  a  straight  line 34 

33.  To  lay  off  the  length  of  a  given  straight 

line  upon  an  arc 34 

34.  To  draw  a  perpendicular   to  a  line, 

from  or  through  a  given  point ...     34 

35.  To  draw  a  line  at  an  angle  of  any 

given  magnitude  in  a  quadrant  with 

a  given  line 35 

36.  To  draw  a  parallel  to  a  given  straight 

line:  (a)  at  a  given  distance;  or 
(b)  through  a  given  point 36 

37.  To  construct  a  triangle:   having  given 

(a)  the  sides;  (b)  a  side  and  angles 
adjacent  to  it;  or  (c)  a  side,  an  adja- 
cent angle,  and  angle  opposite  the  side  36 


38.  To  construct  an  isosceles  triangle  when 

the  base  and  vertex  angle  are  given    36 

39.  To  construct  an  equilateral   triangle 

when  the  altitude  is  given 37 

40.  To  circumscribe  a  circle  about  a  given 

triangle 37 

41.  To  inscribe  a   circle   within  a  given 

triangle 37 

42.  To    inscribe    an    equilateral    triangle 

within  a  circle 37 

43.  To  circumscribe  an  equilateral  triangle 

aboutacircle 37 

44.  To  construct    a    parallelogram  when 

two  sides   and   the   included  angle 
are  given 37 

45.  To  construct  a  square  on  a  given  side    37 

46.  To  inscribe  a  square  within  a  circle. . .     38 

47.  To   construct    a    square   on   a  given 

diagonal 38 

48.  To  circumscribe  a  square  about  a  circle    38 

49.  To  inscribe  a  regular  pentagon  within 

a  circle 38 

50.  To  construct  a  regular  hexagon  on  a 

given  side 38 

51.  To  inscribe  a  regular  hexagon  within 

a  circle .  .  38 


52.  To  construct  a  regular  hexagon  on  a 

given  long  diagonal 38 

53.  To  inscribe  a  regular  octagon  within  a 

circle 39 

54.  To   circumscribe   a   regular   octagon 

about  a  circle *. 39 

55.  To   construct    any    regular   polygon. 

General  methods:  having  given  (a)  a 
side;  (b)  the  circumscribing  circle; 
or  (c)  the  inscribed  circle 39 

56.  To  construct  a  polygon  similar  to  a 

given  polygon,  upon  a  given  side. . .     40 

57.  To  plot  a  figure  similar  to  a  given 

figure  by  means  of  a  base  line  or 
center  line,  and  offsets 40 

58.  To  draw  a  tangent  to  a  circle  through 

a  given  point 41 

59.  To  draw  a  tangent  to  two  given  circles    41 

60.  To  draw  a  circular  curve  of  given 


radius  tangent  to  a  given  circular 
curve  and  to  a  given  straight  line, 
or  to  two  given  circular  curves 42 

61.  To  draw  a  circular  curve  tangent  to 
two  given  straight  lines:  having 
given  (a)  a  point  of  tangency;  or 
(b)  the  radius 42 

62'  To  draw  a  circular  curve  tangent  to 

three  straight  lines 42 

63.  To  draw  circular  curves  tangent  to 

two  given  parallel  straight  lines  and 

to  each  other,  at  a  given  point 43 

64.  To  draw  an  ellipse  when  the  axes  are 

given: 

(a)  By  Focal  Radii 43 

(b)  By  Trammel  Method 43 

(c)  By  Revolution  of  a  Circle 43 

(d)  By  Parallelogram  Method 43 

(e)  By  Circular  Arcs 44 


CHAPTER  V 
ORTHOGRAPHIC   PROJECTION 


65.  General  Principles 45 

66.  To  draw  the  front,  top,  and  side  views 

of  a  rectangular  object 49 

67.  Objects  Having  Surfaces  Oblique  to 

the  Co-ordinate  Planes 51 

68.  Objects  Having  Curved  Surfaces 51 

69.  Projections  upon    Oblique    Auxiliary 

Planes 52 

70.  Revolution  of  Surfaces.  ..  .  54 


71.  True  Length   and  Position  of  Lines 

Oblique  to  the  Co-ordinate  Planes . .     55 

72.  Objects  Oblique  to  the  Co-ordinate 

Planes 56 

73.  Partial   Views,  and  Use  of  Auxiliary 

Views    in     Determining     Required 
Views 58 

74.  Rules  Governing  the  Position  of  Lines 

and  Surfaces  Relative  to  Any  Two 
Perpendicular  Planes  of  Projection.     61 


75.  Principles  a.nd  Methods 

76.  Objects  Having  Plane  Surfaces. 


CHAPTER  VI 
PLANE   SECTIONS 

...     62  77.  Objects  Having  Curved  Surfaces . 

.     62 


t>4 


78.  Principles  and  Methods 

79.  Objects  Having  Plane  Surfaces 


CHAPTER  VII 
INTERSECTION  OF  SURFACES 

80.  Objects  Having  Curved  Surfaces . 


CHAPTER  VIII 
DEVELOPMENT  OF  SURFACES 


81.  Principles  and  Methods 71 

82.  Objects  Having  Plane  Surfaces 72 

83.  Objects    Having   Cylindric  or  Conic 

Surfaces  . .  .73 


84.  Objects    Having    Double   Curved  or 

Warped  Surfaces 77 

85.  Intersecting  Surfaces 78 


CHAPTER  IX 
MECHANICAL   PICTORIAL   DRAWINGS 


86.  Character  and  Purpose  of  the  Drawing  80 

87.  Isometric  Projection 81 

88.  To  draw  the  isometric  of  a  rectangular 

object 81 


89.  To  draw  the  isometric  of  an  object 

involving  non-isometric  figures 83 

90.  Oblique  Projection 86 

91.  Shade  Lines,  Shadow  Lines,  and  Line 

Shading 87 


CHAPTER  X 
WORKING   DRAWINGS 


92.  Character  and  Purpose  of  the  Drawing  89 

93.  Typesof  Drawings 89 

94.  Position  of  Object  and  Arrangement 

of  Views 92 

95.  Selection  and  Number  of  Views,  etc. .  .  92 

96.  Center  Lines 94 

97.  Conventional  Representations 94 

98.  Sectional  Views 96 

99.  Broken  Views..  ,  .  98 


100.  Standard  Sizes  of  Sheets  and  Scale  of 

Drawings 99 

101.  Dimensioning 100 

102.  Lettering 105 

103.  Shadow  Lining  and  Line  Shading.  .  .  108 

104.  Sketching 108 

105.  Making  Scale  Drawings 109 

106.  Tracing  and  Blue-printing 115 

107.  Checking  Drawings 120 

108.  Reading  Drawings 120 


CHAPTER  XI 
HELICAL  CURVES,  THREADED  PARTS,  AND  SPRINGS 


109.  Helices 

110.  Screw  Threads. . 

111.  Pipe  Threads.  .  . 


122 
123 
128 


112.  Bolts... 

113.  Screws.. 

114.  Springs. 


128 
131 
132 


SYMBOLS  AND  GENERAL  ABBREVIATIONS 


||       parallel;   ||s     parallels 

_L     perpendicular;    J_s    perpendiculars 

Z     angle;    Zs     angles 

A     triangle;    As     triangles 

O     circle;  Os     circles 

C.  L.      center  line;   C.  Ls.     center  lines 


diam.  diameter;  diams.     diameters 

hor.  horizontal;  hors.     horizontals 

pt.  point;  pts.    points 

rad.  radius 

st.  straight 

vert.  vertical;  verts,     verticals 


ESSENTIALS   OF 
MECHANICAL    DRAFTING 


CHAPTER   I 
INTRODUCTION 

1.  Nature  and  Uses  of  Mechanical  Drafting.  Mechanical  drafting  or 
mechanical  drawing  is  the  art  of  making  the  conventional  representations  used  by 
engineers,  architects,  and  inventors  in  working  out  and  recording  the  details  of 
their  constructive  designs,  and  the  means  by  which  ideas  of  the  exact  form  or 
shape,  dimension,  and  arrangement  of  parts  in  objects  of  a  structural  charac- 
ter are  universally  expressed  and  made  intelligible  to  others. 

Mechanical  drafting  enables  constructive  work  of  any  kind  to  be  carried  on 
with  accuracy  and  economy  of  time  and  material,  and  takes  the  place  of  lengthy 
verbal  description  which  would  fail  to  express  with  clearness  and  exactness  the 
definite  information  required  by  the  workman. 

It  will  be  seen  from  Fig.  179  that  certain  general  peculiarities  of  the  form  and 
structure  of  an  object  may  be  understood  from  an  ordinary  pictorial  representa- 
tion, but  that  it  cannot  show  the  exact  form,  size,  and  relation  of  all  the  lines 
and  surfaces;  hence  the  necessity  for  mechanical  drawings  which  show  all  hidden 
as  well  as  visible  parts  of  an  object  as  they  are  and  not  as  they  would  appear 
to  the  eye. 

Mechanical  drafting  is  thus  the  graphic  language  of  the  constructive  or 
mechanic  arts,  and  ability  to  read  or  comprehend  mechanical  drawings  is  of  as 
great  importance  to  the  workman,  builder,  and  manufacturer  as  ability  to  make 
such  representations  is  to  the  designer  or  draftsman;  and  a  knowledge  of  general 
drafting  principles  is  of  value  to  almost  all  men  irrespective  of  their  vocations. 

Because  of  the  exact  nature  of  the  facts  which  it  is  intended  to  record  or  convey 
the  drawing  is  generally  executed  with  the  aid  of  instruments. 

The  mechanical  character  of  the  representation,  together  with  its  purpose 
and  the  usual  means  of  execution,  gives  mechanical  drafting  its  name. 

Machine  drafting,  architectural  drafting,  and  engineering  drafting  are  specific 
applications  of  mechanical  drafting. 

A  mechanical  drawing  properly  dimensioned  in  figures  and  prepared  as  a 
guide  in  constructing  the  object  is  called  a  working  drawing. 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


2.  Geometric  Terms,  Definitions,  etc.  Geometry  is  the  science  which 
describes  the  definite  figures  (forms  or  shapes)  upon  which  all  objects  however 
complex  are  based,  and  the  principles  and  methods  by  which  these  figures  may  be 
measured  and  graphically  constructed. 

Geometry  is  thus  fundamental  in  mechanical  drafting  and  in  all  the  con- 
structive arts. 

The  terms  defined  in  this  chapter  are  commonly  involved  in  both. 

(a)  GENERAL  DEFINITIONS.  A  physical  solid  or  material  object  occupies  a  certain  portion  of 
space  and  has  shape,  size,  weight,  color,  etc.  Geometry  is  concerned  simply  with  the  shape  and 
size  of  the  space  which  a  physical  solid  occupies  or  is  conceived  to  occupy;  hence — a  geometric  solid 
is  a  limited  portion  of  space. 

A  solid  has  dimensions  or  extent  in  three  principal  directions  at  right  Zs  to  each  other;  namely, 
length,  breadth  (or  width),  and  thickness  (height,  altitude,  or  depth). 

The  boundaries  of  a  solid  are  called  surfaces. 

A  surface  has  only  two  dimensions:  length  and  breadth  (or  width). 

The  boundaries  of  a  surface  are  called  lines. 

A  line  has  only  one  dimension :  length. 

The  limits  or  ends  of  a  line  are  called  points. 

A  point  has  position  but  no  dimension. 

Points,  lines,  and  surfaces  may  be  considered  as  apart  from  a  solid,  or  as  combined  in  any  con- 
ceivable figure;  also  a  line  may  be  imagined  as  generated  by  a  point,  a  surface  by  a  line,  and  a  solid 
by  a  surface,  in  motion. 

Similar  figures  are  those  having  the  same  shape;  equivalent  figures  those  having  the  same  size; 
and  equal  or  congruent  figures,  those  having  the  same  shape  and  size. 

A  figure  that  lies  wholly  in  one  plane  is  a  plane  figure.  (See  Art.  (h).)  A  figure  whose  lires  are 
straight  is  rectilinear;  one  whose  lines  are  curved  is  curvilinear. 

The  axis  of  a  figure  is  a  st.  line  which  passes  through  its  center,  and  about  which  it  is  symmetrical 
or  balanced. 

An  axis  of  revolution  is  a  st.  line  about  which  a  figure  is  revolved. 

When  two  lines,  two  surfaces,  or  a  line  and  a  surface  meet  or  cross  they  are  said  to  intersect  or 
cut  each  other  and  the  pt.  or  line  in  which  they  intersect  is  their  intersection. 

A  bisector  is  a  pt.,  line,  or  plane  which  divides  a  figure  into 

• FlG.  I         two  equal  parts,  that  is,  bisects  it.     To  trisect  is  to  divide  into 

three  equal  parts;  to  quadrisect,  into  four  equal  parts. 


(b)  LINES.  A  straight  or  right  line  has  the  same  direction 
throughout.  Fig.  1. 

A  curved  line  or  curve  is  one  no  part  of  which  is  straight. 
Fig.  2. 

A  reversed  curve  is  one  whose  direction  of  curvature  changes. 
Fig.  3. 

A  horizontal  line  is  one  that  is  level  throughout.  In  draw- 
ing, the  term  is  applied  to  a  st.  line  drawn  from  left  to  right, 
without  slant.  Fig.  1. 

A  vertical  line  is  one  that  is  upright  or  plumb.  In  drawing, 
the  term  is  applied  to  a  st.  line  drawn  from  bottom  to  top,  with- 
out slant.  Fig.  4. 

An  oblique  line  is  one  that  slants.     Fig.  5. 

Two  lines  having  the  same  relative  direction  are  parallel  to 
each  other.  They  are  the  same  distance  apart  throughout. 
Fig.  6. 


FlG. 6 


INTRODUCTION 


3 


Two  st.  lines  which  extend  from  the  same  pt.  or  which  would  intersect  if  extended,  are  said  to  be 
at  an  angle  with  each  other  (Fig.  7),  and  are  perpendicular,  or  oblique,  to  each  other  according  as 
the  included  Z  is  a  right  Z  or  an  oblique  Z.  See  Art.  (c). 

Two  curves,  or  a  st.  line  and  a  curve,  are  tangent  to  each  other  when  they  touch  in  but  one  pt. 
and  cannot  intersect.  The  pt.  is  the  point  of  tangency.  Figs.  11  (a),  12,  14. 

An  ordinate  or  offset  is  the  _|_  distance  of  a  pt.  from  a  given  st.  line,  or  plane,  of  reference.  A-l, 
B-2,  etc.,  Fig.  103. 

Co-ordinates  are  ordinates  of  the  same  pt.,  measured  ||  to  two,  or  three,  mutually  J_  lines,  or 
planes,  of  reference.  C-O,  C-O,  (X>,  Fig.  118. 


(a) 


\ 


FIG.  7 


FIG.  9 


(c)  ANGLES.     These  definitions  refer  to  plane  Zs  only. 

An  angle  is  the  opening  between  two  st.  lines,  called  the  sides  of  the  angle,  which  extend  from 
a  pt.  called  the  vertex.  BAG,  Fig.  7.  An  angle  may  be  considered  as  generated  by  a  st.  line  revolved 
about  one  of  its  ends. 

The  size  of  an  angle  depends  upon  the  relative  direction  of  the  sides  and  not  upon  their  length. 
When  the  sides  extend  in  opposite  directions,  so  as  to  lie  in  the  same  st.  line,  the  Z  is  a  straight  angle. 
When  the  directions  are  such  that  the  Zs  formed  by  extending  the  sides  beyond  the  vertex  are 
equal,  each  Z  is  a  right  angle.  A  right  Z  is  equal  to  half  a  st.  Z .  Fig.  8. 

An  Z  less  than  a  right  Z  is  an  acute  angle  (Fig.  7) ;  one  greater  than  a  right  Z  and  less  than  a 
st.  Z  is  an  obtuse  angle  (Fig.  9);  one  greater  than  a  st.  Z  and  less  than  two  st.  Zs  is  a  reflex  angle. 
Note  that  two  st.  lines  extending  from  a  pt.  always  form  two  Zs,  as  Za  and  Zb,  Fig.  9. 

Angles  other  than  right  and  straight  Z  s  are  oblique  angles. 

Two  Zs  having  the  same  vertex  and  a  common  side  between  them  are  adjacent  angles.  DCF 
and  FCE,  Fig.  10. 

When  two  st.  lines  intersect,  the  opposite  or  non-adjacent  Zs  are  equal  and  are  called  vertical 
angles.  GCD  and  FCE,  also  Zs  DCF  and  ECG,  Fig.  10. 


FIG. 10 


An  angle  is  said  to  be  measured  by  an  arc,  described  from  its  vertex  as  center  and  included  by  its 
sides.  The  unit  of  measure  is  an  angle  whose  arc  is  a  degree  (s^  of  a  circumference).  (See  Art.  (d).) 
Thus  a  st.  Z  is  one  of  180°,  a  right  Z  one  of  90°,  and  the  whole  angular  space  about  a  pt.  in  a  plane 
equals  360°.  See  Fig.  10. 

Lines  can  be  drawn  in  four  directions  from  a  given  pt.,  at  the  same  given  Z  with  a  given  line; 
thus  in  Fig.  10,  C-D,  C-E,  C-F,  and  C-G  each  make  an  Z  of  30°  with  A-A.  Each  of  these  lines  makes 
two  Zs  with  the  hor.  A-A  and  two  with  the  vert.  B-B.  Thus  C-D  makes  Zs  of  150°  and  30°  with 
A-A,  and  60°  and  120°  with  B-B. 


4  ESSENTIALS  OF  MECHANICAL  DRAFTING 

In  speaking  of  the  /.&  formed  by  two  lines,  the  lesser  is  the  one  usually  named,  and,  unless  other- 
wise stated,  a  line  at  15°,  30°,  45°,  etc.,  is  understood  to  mean  15°,  30°,  45°,  etc.,  with  the  hor.  direction. 

(d)  CURVILINEAR  FIGURES.  A  circle*  is  a  plane  figure  bounded  by  a  curve  called  its  circumference, 
all  pts.  of  which  are  equidistant  from  a  pt.  within  called  the  center.  Fig.  11  (a). 

Any  part  of  a  circumference  is  an  arc. 

A  st.  line  intersecting  a  circular  curve  in  two  pts.  is  a  secant,  F-G.  A  st.  line  joining  two  pts. 
in  the  curve  is  a  chord,  H-I,  A-B. 

A  chord  through  the  center  is  a  diameter. 

A  straight  line  from  the  center  to  the  curve  is  a  radius,  C-A,  C-D,  C-E. 

A  segment  is  a  portion  of  a  O  bounded  by  an  arc  and  its  chord. 

A  segment  equal  to  half  a  O  is  a  semicircle*. 

A  sector  is  a  portion  of  a  O  bounded  by  an  arc  and  two  radii. 

A  sector  equal  to  one-fourth  of  a  O  is  a  quadrant*. 

An  Z  formed  by  two  chords  from  the  same  pt.  is  an  inscribed  angle.  Fig.  1 1  (b) .  An  angle  is 
inscribed  in  a  segment  when  its  sides  join  the  ends  of  the  arc. 

The  Z  included  by  two  radii  is  a  central  angle. 


FIG.  12 


x (b) 


C   J 


FlG.    14 


The  circumference  of  a  O  ia  conceived  to  contain  360  equal  parts  called  degrees  (360°),  each  degree 
60  equal  parts  called  minutes  (60'),  and  each  minute  60  equal  parts  called  seconds  (60"). 

A  st.  tangent  is  ±  to  a  rad.  at  the  pt.  of  tangency.  J-K,  Fig.  11  (a).  The  pt.  of  tangency  of  two 
circular  curves  is  in  their  line  of  centers.  Fig.  12. 

Two  circles  or  arcs  having  the  same  center  are  concentric.     Fig.  13(a). 

Two  circles  not  having  the  same  center  are  eccentric  when  one  is  within  the  other.     Fig.  13(b). 

An  ellipse*  is  a  plane  figure  bounded  by  a  curve  called  its  circumference,  the  sum  of  the  distances 
of  every  pt.  of  which,  from  two  pts.  within,  called  the  focuses  or  foci,  is  constant.  Fig.  14. 

A  pt.  midway  between  the  foci  is  called  the  center. 

A  st.  line  joining  any  two  pts.  in  the  curve  is  a  chord. 

A  chord  through  the  center  is  a  diameter. 

A  diam.  containing  the  foci  and  center  is  the  long  or  major  axis.  A  diam.  J_  to  the  major  axis 
is  the  short  or  minor  axis.  The  major  and  minor  axes  are  the  longest  and  shortest  diameters  of  the 
ellipse  and  bisect  each  other  at  the  center. 

A  st.  line  from  either  focus  to  any  pt.  in  the  curve  is  a  focal  radius. 

The  sum  of  the  focal  radii  of  any  pt.  is  equal  to  the  major  axis. 

A  st.  tangent  bisects  the  Z  between  one  focal  rad.  and  the  other  extended  at  the  pt.  of  tangency. 
When  one  diam.  is  ||  to  the  tangents  at  the  ends  of  another,  the  diams.  are  conjugate  to  each  other. 


*The  terms  "  circle."  "  semicircle,"  "  quadrant,  "  and  "  ellipse  "  are  also  used  to  denote  merely  the  curve. 


INTRODUCTION 


(e)  POLYGONS.     These  definitions  refer  to  plane  polygons  only. 

A  polygon  is  a  plane  figure  bounded  by  st.  lines  called  its  sides.  The  sum  of  the  sides  is  the 
perimeter;  the  Zs  formed  by  the  sides  are  the  angles;  and  the  vertices  of  the  Zs,  the  vertices  of  the 
polygon.  Figs.  15-30. 

A  polygon  is  equilateral  when  all  of  its  sides  are  equal  (Figs.  15,  24,  27) ;  equiangular  when  all  its 
Zs  are  equal  (Fig.  21);  regular  when  both  equilateral  and  equiangular  (Figs.  15,22,  28-30) ; otherwise, 
it  is  irregular. 

The  base  of  a  polygon  is  the  side  upon  which  it  is  supposed  to  rest.  In  general  any  side  may  be 
considered  as  the  base. 

The  altitude  is  the  _|_  distance  between  the  base,  or  base  extended,  and  the  farthermost  vertex 
or  side.  A-B  in  Figs.  15,  17,  19,  23,  26. 

A  diagonal  is  a  st.  line  joining  any  two  non-consecutive 
vertices.  A-B,  Fig.  21;  A-B  and  A-C,  Fig.  29. 

A  polygon  is  inscribed  in  a  O  when  all  its  sides  are  chords 
of  the  O,  and  circumscribed  about  a  O  when  all  its  sides  are 
tangents  of  the  O.  Also  the  O  is  circumscribed  about  the 
inscribed  polygon  and  inscribed  in  the  circumscribed  polygon. 
Fig.  28. 

The  center  of  a  regular  polygon  is  the  center  of  the  inscribed 
or  circumscribed  O . 

The  rad.  of  the  inscribed  O  is  the  apothem,  and  the  rad.  of 
the  circumscribed  O  the  radius  of  the  polygon.  Fig.  28. 

In  a  regular  polygon  of  an  even  number  of  sides  the  diameter 
of  the  inscribed  O  is  often  called  the  short  diameter,  and  that  of 
the  circumscribed  Q  the  long  diameter  of  the  polygon. 

The  sum  of  the  Zs  of  any  polygon  is  equal  to  two  right 
Zs  (180°),  taken  as  many  times  less  two  as  the  figure  has  sides. 
See  Fig.  20. 

The  Z  included  by  the  radii  to  the  ends  of  any  side  of 
a  regular  polygon  is  called  the  angle  at  the  center.  It  is  equal  to 
360°  divided  by  the  number  of  sides.  Fig.  30. 

A  polygon  of  three  sides  is  a  triangle;  of  four,  a  quadrilateral;  Pi  G .  19 

of  five,  a  pentagon;  of  six,  a  hexagon;  of  seven,  a  heptagon;  of 
eight,  an  octagon;   of  nine,  a  nonagon;   of  ten,  a  decagon. 

(a) 


16 


(b) 


FIG.  20 


(f)  TRIANGLES.    Triangles  are  classified  according  to  relative 
length  of  sides;   as  equilateral,  isosceles,  and  scalene. 

An  equilateral   triangle  has  all  sides  equal;  it  is  also  equi- 
angular.    Fig.  15. 

An  isosceles  triangle  has  two  sides  equal;  two    Zs  are  also 
equal.     Fig.  16.     The  equal  sides  are  usually  called  the  sides,  and  the  other  side,  the  base. 

A  scalene  triangle  has  no  two  sides  equal;  its  Zs  are  also  unequal.     Fig.  17. 

Triangles  are  classified  according  to  kind  of  Zs;  as  right,  obtuse,  and  acute. 

A  A  is  a  right  triangle  when  one   Z  is  a  right   Z.     Fig.  18.     The  side  opposite  the  right   Z  is 
called  the  hypotenuse,  the  others  are  usually  called  the  sides. 

A  A  is  an  obtuse  triangle  when  one  Z  is  obtuse.     Fig.  19. 

A  A  is  an  acute  triangle  when  all  Zs  are  acute.     Fig.  20 (a). 

The  Z  opposite  the  base  of  a  A  is  called  the  vertex  angle,  and  its  vertex ,  the  vertex  of  the  triangle. 

(g)  QUADRILATERALS.      A  parallelogram  is  a  quadrilateral  whose  opposite  sides  are  ||.     Figs. 
21-24. 

A  rectangle  is  a  right-angled  parallelogram.     Figs.  21,  22. 

A  square  is  an  equilateral  rectangle.     Fig.  22. 

A  rectangle  whose  opposite  sides  only  are  equal  is  often  called  an  oblong.     Fig.  21. 

A  rhomboid  is  an  oblique-angled  parallelogram.     Figs.  23,  24. 


KSSKNTIALS  <>F   M  IX'll  AM( '  A  I.   DHAFTINC 


FIG.  27 


FIG.  25 


FIG  30 


A  rhombus  is  an  equilateral  rhomboid.     Fig.  24. 

The  side  | 1  to  the  base  of  a  parallelogram  is  called  the  upper  base,  the  other  is  the  lower  base. 
A  trapezoid  is  a  quadrilateral  which  has  two  sides  only  ||.     Fig.  25.     The  parallel  sides  are  the 
upper  and  lower  bases. 

A  trapezium  is  a  quadrilateral  which  has  no  two  sides    .     Fig.  26. 

(h)  SURFACES.  A  plane  surface  or  plane  is  a  surface  such  that  a  st.  line  through  any  two  pts.  in 
it  lies  wholly  in  the  surface. 

A  curved  surface  is  one  no  part  of  which  is  plane.  If  a  semicircular  arc  be  revolved  about  its 
chord,  it  will  generate  a  spheric  surface.  Fig.  46. 

A  st.  line  which  moves  1 1  to  its  first  position  and  constantly  touches  a  fixed  curve  not  in  the  plane 
of  the  line,  will  generate  a  cylindric  surface.  Fig.  39. 

A  moving  st.  line  which  constantly  intersects  a  fixed  curve  and  passes  through  a  fixed  pt.  not  in 
the  plane  of  the  curve  will  generate  a  conic  surface.  The  fixed  pt.  is  its  vertex.  Fig.  43. 

Curved  surfaces  generated  by  st.  lines  are  single  curved  surfaces.  The  generating  line  in  any  of 
its  positions  is  called  an  element. 

A  warped  surface  is  one  of  single  curvature  in  which  no  two  consecutive  elements  are  1 1  or  inter- 
secting. A  in  Fig.  158. 

A  double  curved  surface  is  one  generated  by  a  curve  of  which  no  two  consecutive  positions  are 
||;  as  the  surface  of  a  sphere,  ellipsoid,  etc.  Figs.  46,  47. 

A  curved  surface  generated  by  the  revolution  of  a  line  about  an  axis  is  called  a  surface  of  revolution; 
as  a  right  circular  cylindric  or  conic  surface,  etc.  Figs.  39,  43,  46,  47.  Each  pt.  of  the  generating 
line  describes  a  O  whose  plane  is  J_  to  the  axis. 

A  plane  surface  is  horizontal  when  it  is  level  throughout;  vertical  when  _L  to  a  hor.  plane;  oblique 
when  neither  hor.  nor  vert. 

Two  surfaces^  or  a  line  and  a  surface,  are  parallel  when  they  are  the  same  distance  apart  throughout. 

Two  plane  surfaces  which  extend  from  the  same  st.  line  or  which  would  intersect  if  extended,  are 
said  to  be  at  an  angle  with  each  other;  and  are  perpendicular,  or  oblique,  to  each  other  according  as 
the  included  Z  is  a  right  or  an  oblique  dihedral. 

A  dihedral  or  dihedral  angle  is  the  opening  between  two  planes  called  the /aces  of  the  angle,  which 
intersect  in  a  st.  line  called  its  edge. 

A  dihedral  is  measured  by  the  plane  Z  formed  by  two  st.  lines,  one  in  each  face  and  J_  to  the 
edge  at  the  same  pt. 


INTRODUCTION  7 

A  dihedral  is  right,  acute,  or  obtuse  according  as  its  measure  is  a'right,  acute,  or  obtuse  plane  Z. 

A  st.  line  and  a  plane  are  J_  to  each  other  when  the  line  is  J_  to  every  st.  line  through  its  foot 
or  pt.  of  intersection  with  the  plane. 

The  Z  a  line  makes  with  a  plane  is  the  Z  which  it  makes  with  a  line  in  the  plane  passing 
through  its  foot  and  the  foot  of  a  _l_  from  any  other  pt.  in  the  line. 

A  plane  and  a  curved  surface,  or  two  curved  surfaces,  are  tangent  when  they  touch  in  but  one  pt. 
or  in  one  line,  and  cannot  intersect.  The  pt.  or  line  is  the  point  or  line  of  tangency. 

A  st.  line  or  curve  and  a  curved  surface,  or  a  curve  and  plane,  are  tangent  when  they  touch  in 
but  one  pt.  and  cannot  intersect. 

(i)  SOLIDS.     The  base  is  the  plane  surface  of  the  solid  upon  which  it  is  supposed  to  rest. 

The  altitude  is  the  _|_  distance  between  the  plane  of  the  base  and  the  farthermost  vertex  or  part. 

A  plinth  is  a  prism,  or  cylinder,  whose  altitude  is  its  least  dimension.     Figs.  33,  42. 

A  plane  section  is  the  figure  formed  by  the  intersection  of  a  solid  with  a  plane  passing  through  it. 

A  solid  of  revolution  is  one  which  may  be  generated  by  a  plane  surface  revolving  about  an  axis. 

A  solid  bounded  by  plane  surfaces  is  called  a  polyhedron.  Figs.  31-38.  The  bounding  surfaces 
are  its  faces;  the  intersection  of  its  faces,  the  edges;  and  the  vertices  of  the  faces,  the  vertices  of  the 
polyhedron. 

A  diagonal  of  a  polyhedron  is  a  st.  line  joining  any  two  vertices  not  in  the  same  face,  as  A-B,  Fig. 
31  (a). 


FIG. 31 


FIG.  32 


FIG.  33 


FIG.  35 


(j)  PRISMS.  A  prism  is  a  polyhedron  bounded  by  two  equal  polygons  called  its  bases,  and  by 
three  or  more  parallelograms  called  its  lateral  faces.  The  intersections  of  its  lateral  faces  are  its 
lateral  edges;  the  others  are  its  base  edges.  Figs.  31,  32,  33,  35. 

Prisms  are  named  from  their  bases;  as  triangular,  square,  etc. 

A  prism  whose  base  centers  lie  in  a  _L  to  its  bases  is  a  right  prism.  Figs.  31-33.  All  others  are 
oblique.  Fig.  35. 

A  regular  prism  is  a  right  prism  whose  bases  are  regular  polygons.  Its  lateral  faces  are  equal 
rectangles. 

A  cube  is  a  regular  prism  whose  six  faces  are  equal  squares.     Fig.  32. 

A  right  section  of  a  prism  is  a  section  _L  to  its  lateral  edges. 

A  truncated  prism  is  the  portion  of  a  prism  included  between  a  base  and  a  section  oblique  to  the 
base.  Fig.  34. 

(k)  PYRAMIDS.  A  pyramid  is  a  polyhedron  bounded  by  a  polygon  called  its  base,  and  three  or 
more  As  called  its  lateral  faces,  meeting  in  a  common  pt.  called  the  vertex  of  the  pyramid. 

The  intersection  of  its  lateral  faces  are  its  lateral  edges;  the  others  are  its  base  edges.     Figs.  36,  37. 

Pyramids  are  named  from  their  bases;  as  triangular,  square,  etc. 

A  pyramid  whose  vertex  lies  in  a  J_  to  the  center  of  its  base  is  a  right  pyramid.  Fig.  36.  All 
others  are  oblique.  Fig.  37. 

A  regular  pyramid  is  a  right  pyramid  whose  base  is  a  regular  polygon.  Its  lateral  faces  are 
equal  As. 

The  altitude  of  a  lateral  face  of  angular  pyramid  is  the  slant  height  of  the  pyramid.  C-B,  Fig.  36. 

A  truncated  pyramid  is  the  portion  of  a  pyramid  included  between  the  base  and  any  plane  section. 
Figs.  38  (a)  and  (b).  When  the  section  is  ||  to  the  base,  the  included  portion  is  afrustum  of  a  pyramid. 
Fig.  38(a). 


8  ESSENTIALS  OF  MECHANICAL  DRAFTING 

(1)  CYLINDERS.  A  cylinder  is  a  solid  bounded  by  a  closed  cylindric  surface  called  the  lateral 
surface,  and  two  ||  plane  surfaces  called  its  bases.  Figs.  39,  40,  42. 

A  cylinder  is  named  from  its  bases;  as  circular,  elliptic,  etc.  The  terms  "right,"  "oblique,"  and 
"truncated"  apply  to  a  cylinder  as  to  a  prism. 


FlG.  38 


FIG.  41 


FIG.  42 


A  right  circular  cylinder  may  be  generated  by  the  revolution  of  a  rectangle  about  one  of  its  sides. 
Fig.  39. 

A  right  section  of  a  cylinder  is  a  section  _L  to  its  elements. 

(m)  CONES.  A  cone  is  a  solid  bounded  by  a  closed  conic  surface  called  the  lateral  surface,  and 
a  plane  surface  called  its  base.  Figs.  43,  44. 

A  cone  is  named  from  its  base;  as  circular,  elliptic,  etc.  The  terms  "right"  "oblique,"  "truncated," 
and  "frustum"  apply  to  a  cone  as  to  a  pyramid. 

A  right  circular  cone  may  be  generated  by  the  revolution  of  a  right  A  about  one  of  its  sides.  Fig. 
43.  The  length  of  an  element  of  a  right  circular  cone  is  the  slant  height. 

(n)  A  SPHERE  is  a  solid  bounded  by  a  closed  spheric  surface  every  point  of  which  is  equidistant 
from  a  pt.  within  called  the  center.  Fig.  46.  A  st.  line  from  the  center  to  the  surface  is  a  radius; 
a  st.  line  through  the  center  and  terminated  at  each  end  by  the  surface  is  a  diameter. 


FIG.  43 


FIG.  47 


A  sphere  may  be  generated  by  the  revolution  of  a  semicircle  about  its  diam. 

A  plane  section  through  the  center  is  a  great  circle  of  the  sphere.  Any  others  are  small  circles  of 
the  sphere. 

Any  great  O  divides  a  sphere  into  two  equal  parts  called  hemispheres. 

A  spheroid  (ellipsoid)  is  a  solid  which  may  be  generated  by  the  revolution  of  an  ellipse  about 
either  its  long  or  short  diam.  Fig.  47. 


INTRODUCTION  9 

(o)  AREAS  AND  VOLUMES.  The  number  of  times  a  geometric  magnitude  contains  a  given  unit 
of  measure  of  the  same  kind  is  the  numeric  measure  of  the  magnitude. 

The  ratio  of  two  magnitudes  is  the  quotient  of  their  numeric  measures,  expressed  in  terms  of  the 
same  unit.  Thus  the  ratio  of  2"  to  3"  is  f ,  or  2  :  3. 

The  expression  of  the  equality  of  two  ratios  is  called  a  proportion.  As  f=£.  Read  2  is  to  3 
as  4  is  to  6.  The  quantities  compared  are  said  to  be  in,  proportion  or  proportional,  and  are  called 
the  terms.  The  first  and  last  terms  are  the  extremes  and  the  middle  terms,  the  means.  In  any 
proportion  the  product  of  the  extremes  equals  the  product  of  the  means;  hence,  if  three  terms  are 
given,  the  fourth  may  be  found. 

The  area  of  a  surface  is  its  measure  expressed  in  some  unit  of  surface,  as  a  square  inch,  square 
foot,  etc. 

The  area  (A)  of  a  parallelogram  is  equal  to  the  product  of  its  base  (b)  and  altitude  (a).      A  =  ba. 

The  area  of  a  square  is  equal  to  the  square  of  one  of  its  sides  (s).     A  =  s2. 

The  length  of  the  side  is  equal  to  the  square  root  of  the  area.     s=  I/A. 

The  length  of  the  diagonal  (d)  is  equal  to  the  square  root  of  2  times  the  square  of  the  side. 
d=i/2Xs2  or  d  =  sl/2l  (1/2==  1.414). 

The  area  (A)  of  a  A  is  equal  to  £  the  product  of  its   base  (b)  and  altitude  (a).     A=  £  ba. 

The  altitude  (a)  of  an  equilateral  A  is  equal  to  £  the  product  of  the  side  (s)  and  the  square  root 

of  3.     a-2^p        (1/3=  1.732). 

The  hypotenuse  (h)  of  a  right  A  is  equal  to  the  square  root  of  the  sum  of  the  squares  of  the  other 
two  sides  (b)  and  (c).  h  =  l/b2  +  c2. 

The  area  of  a  trapezoid  is  equal  to  ^  the  product  of  its  altitude  (a)  and  the  sum  of  its  bases  (b) 
and  (b').  A=Ja(b+b'). 

Any  polygon  may  be  divided  into  As.  The  area  of  the  polygon  is  equal  to  the  sum  of  the  areas 
of  its  As. 

The  length  of  the  circumference  (c)  of  a  O  is  equal  (very  nearly)  to  3.1416  times  the  length  of 
the  diameter  (d).  3.1416  is  designated  by  the  Greek  letter  (pi),  c  =  Trd  or  27ir. 

The  area  of  a  O  is  equal  to  TT  times  the  square  of  its  rad.  (r).     A  =  Tir2. 

The  volume  of  a  solid  is  its  measure  expressed  in  some  unit  of  volume,  as  a  cubic  inch,  cubic  foot, 
etc. 

The  volume  (V)  of  a  cube  is  equal  to  the  cube  of  its  edge  (s)  or  third  power  of  its  dimension. 
V  =  s3. 

The  volume  of  a  prism,  or  cylinder,  is  equal  to  the  product  of  its  base  (b)  and  altitude  (a). 
V  =  ba. 

The  lateral  area  (1)  of  a  prism,  or  cylinder,  is  equal  to  the  product  of  a  lateral  edge  or  element 
(e),  and  the  perimeter  (p)  or  circumference  (c),  of  a  right  section.  1  =  ep,  or  1  =  ec. 

The  volume  of  a  pyramid,  or  cone,  is  equal  to  £  the  product  of  its  base  and  altitude.     V  =^ba. 

The  lateral  area  of  a  regular  pyramid,  or  right  circular  cone,  is  equal  to  £  the  product  of  its  slant 
height  (s)  and  the  perimeter,  or  circumference,  of  its  base.  1  =  £sp,  or  1  ==  |sc  =  ?rrs. 

The  area  of  the  surface  (s)  of  a  sphere  is  equal  to  the  product  of  the  circumference  of  a  great 
O  and  its  diam.  (d),  that  is,  2?rrd,  and  is  equivalent  to  the  area  of  4  great  Qs.  s  =  4.irr2. 

The  volume  of  a  sphere  is  equal  to  the  product  of  ^  of  its  rad.  and  the  area  of  its  surface,  that  is, 
|r  X  47H-2  =  |7rr3.  V  =  f  Tir3  or  -jTrd3. 


II) 


KSSKNTIALS  <  )|     MKCIIAMC.M.   DKAI  TINT, 


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INTRODUCTION  11 

3.  General  Instructions  for  Working  Out  Problems.  In  the  solution  of 
graphic  problems  clear  mental  images  of  the  forms  to  be  represented,  definite 
ideas  of  the  purpose  of  the  drawings,  and  the  orderly  application  of  appropriate 
principles  and  working  methods,  are  fundamental. 

Habits  of  accuracy,  thoroughness,  and  neatness  should  be  cultivated  from  the 
outset  as  the  essentials  of  good  workmanship.  The  value  of  the  work  lies  not 
in  the  completed  drawings,  but  in  the  knowledge  and  ability  acquired  by  the 
student  through  his  own  efforts  in  solving  problems  and  in  striving  to  attain 
mastery  of  the  principles  and  methods  which  will  enable  him  to  represent  any 
form  whether  real  or  imaginary. 

Upon  the  presentation  of  a  problem  the  student  should  first  form  a 
definite  idea  of  what  is  required,  the  conditions  and  principles  involved,  and  the 
method  of  construction  to  be  employed.  He  should  then  start  the  drawing, 
beginning  with  the  parts  that  are  known  or  given.  These  will  suggest  other 
parts.  It  is  not  necessary  to  imagine  the  complete  solution  before  beginning 
to  draw. 

Test  and  correct  each  stage  of  the  solution  before  proceeding  with  the  next. 
Upon  the  completion  the  student  should  make  a  brief  notebook  summary  for 
future  reference. 

For  use  of  instruments  and  materials,  see  Chap.  II. 

For  general  instructions  in  penciling  and  finish  rendering,  see  Chap.  III. 

(a)  GEOMETRIC     CONSTRUCTION     SHEETS.     Chap.   IV.       Unless    otherwise 
directed  solve  problems  by  practical  methods  whenever  such  are  known  or  given. 
Within  reasonable  limits  prove  or  test  the  constructions  by  geometric  methods, 
using  compasses  and   one  triangle  only;   and  retain  all  working  lines.     Make 
constructions  as  large  as  practicable  to  insure  greater  accuracy,  and  extend  the 
working  lines  beyond  the  pts.  as  shown  in  the  figures. 

To  avoid  impairing  the  accuracy  of  the  constructions,  it  is  recommended  that 
they  be  left  in  pencil. 

(b)  ORTHOGRAPHIC  PROJECTION  SHEETS.   Chaps.  V-VIII.    Locate  first  the  traces 
of  the  planes,  or  equivalent  lines  of  reference  (base  or  C.  Ls.).      Then  proceed 
to  determine  the  views,  beginning  in  general  with  that  view  or  part  about  which 
most  is  known;  in  the  case  of  an  object,  for  example,  the  view  which  will  show 
the  largest  number  of  the  lines  and  surfaces  of  the  object  in  their  exact  form  and 
dimension.     Instead  of  completing  the  views  separately  it  is  usually  desirable 
to  carry  along  the  views  of  corresponding  parts  at  the  same  time.     Unless  other- 
wise directed  use  practical  rather  than  geometric  methods  for  the  constructions, 
and  obtain  the  solutions  by  the  aid  of  dimensioned  freehand  sketches  from  objects. 
See  Art.  104. 

To  determine  the  positions  of  the  lines  and  surfaces  more  readily,  number  or 
letter  each  pt.  lightly  as  located,  marking  the  corresponding  views  to  indicate 
that  they  represent  the  same  pt.  of  the  object.  In  locating  positions  of  centers 
or  other  important  pts.,  small  freehand  Os  may  be  penciled  about  them.  When 
the  drawing  has  been  approved,  these  notation  marks  and  Os  should  be  erased. 


12  ESSENTIALS  OF  MECHANICAL  DKAFTINd 

(c)  ISOMETRIC  AND  OBLIQUE  PROJECTION  SHEETS.     Chap.  IX.     Locate  first 
the  axes,  or  reference  lines,  then  locate  the  main  lines  or  surfaces  of  the  object. 
Gradually  work  from  these  to  the  more  important  details,  then  proceed  to  the 
smaller  details. 

Obtain  the  dimensions  from  objects;  exact  orthographic  views;  working 
drawings;  or  from  dimensioned  sketches  in  orthographic,  isometric,  or  oblique 
projection,  as  may  be  directed. 

(d)  WORKING  DRAWINGS.      Chap.  X.      Locate  first  the  main  C.Ls.  of  the 
views  according  to  the  layout  sketch  (Art.  105  (a)),  then  locate  the  main  lines  or 
surfaces  of  the  object,  as  indicated  in  Fig.  209  (a),  beginning  with  the  view  about 
which  most  is  known,  as  in  Art.  (b). 

Proceed  in  like  manner  with  the  more  important  modifications  or  details 
of  the  main  body  (Fig.  209  (b))and  from  these  work  down  to  the  smaller  details. 
Next,  add  the  dimension  and  extension  lines,  arrowheads,  figures,  and  lettering. 
Fig.  209  (c),  (d).  Finally  indicate  the  section  lines.  Do  not  try  to  complete 
the  views  separately.  Within  practical  limits  those  of  corresponding  parts 
should  be  carried  along  at  the  same  time.  Use  practical  methods  for  the  con- 
structions whenever  adequate. 

Obtain  the  working  data  from  dimensioned  sketches  made  from  objects,  or 
as  directed. 

For  methods  of  representing  the  commoner  forms  of  bolts,  screws,  etc.,  see 
Chap.  XI.  For  tables  of  sizes,  and  the  construction  and  proportions  of  other 
machine  details,  see  manufacturers'  catalogs,  books  on  machine  design,  and 
engineers'  handbooks.  For  sizes,  etc.,  of  details  of  wood  construction,  see  books 
on  joinery,  etc. 


CHAPTER   II 
INSTRUMENTS,    MATERIALS,   AND   THEIR   USE 

4.  List  of  Equipment.     The  instruments  and  materials  ordinarily  needed 
are  enumerated  in  the  following.     A  good  equipment  is  indispensable  to  good 
work.     If  not  furnished  by  the  school,  its  selection  should  be  intrusted  to  an 
experienced  draftsman. 

Set  of  Instruments,  consisting  of  Compasses  5f"  (joint  in  both  legs  preferred),  with  Lead  Holder, 
Pen,  and  Lengthening  Bar  attachments;  Ruling  Pen,  medium;  Dividers,  5";  Bow  Pencil;  Bow 
Pen;  and  Bow  Dividers. 

Leads,  grade  4H,  for  compasses  and  bow  pencil. 

Drawing  Board,  18"  x  24",  is  suitable  for  most  work. 

T-square,  fixed  head,  blade  slightly  longer  than  board. 

Triangles,  one  45°,  7",  and  one  30°  x  60°,  9"  (celluloid  preferred). 

Scale,  12",  flat,  both  edges  divided  into  full  size  inches,  halves,  4ths,  8ths,  and  16ths,  and  the 
first  inch,  or  first  and  last,  into  32ds. 

Or,  a  12"  architect's  triangular  scale:  one  edge  divided  into  full  size  inches  and  fractions,  to 
16ths,  and  the  others  to  scales  of  3",  1£",  1",  f",  f",  f",  i",  ^",  £",  and  ^"  to  1  foot. 

(The  flat  scale  is  recommended  if  scales  of  full,  hah",  quarter,  eighth,  and  sixteenth  sizes  only 
are  to  be  used.) 

Curve  Rulers,  one  or  two,  similar  to  those  shown  in  Figs.  66,  67  (celluloid  preferred). 

Drawing  Pencils,  one  hard,  grade  4H,  5H,  or  6H,  and  one  medium  hard,  grade  H  or  2H. 

Erasers,  one  soft  rubber  for  pencil  erasing  and  one  hard  rubber  for  ink  erasing. 

Needle-point,  a  fine  needle  inserted  in  a  wooden  handle  about  3|"-long,  for  use  in  fixing  pts. 

Tacks,  1  oz.  copper,  or  small  thumb-tacks,  for  fastening  paper  to  board. 

Tack-driver.  A  small  screw-driver  ground  to  a  thin  edge  and  slightly  bent  at  the  end  will  be 
suitable. 

Knife,  for  pencil  sharpening  (one  with  broad  blade  preferred). 

Lead-pointer,  a  strip  of  No.  \  sandpaper  or  No.  120  emery  cloth,  about  f "  x  4",  glued  upon 
a  flat  strip  of  wood. 

Penholder  and  Writing  Pens,  for  lettering,  etc.  (Tapering  handle  with  cork  grip  preferred.) 
Pens  should  be  medium  and  coarse. 

Penwiper,  a  piece  of  cloth  or  wash  leather. 

Drawing  Ink,  one  bottle  waterproof  black;  writing  ink  is  unsuitable. 

Drawing  Paper,  hard  and  tough,  with  a  surface  not  easily  roughened  by  erasures.  For  sketches 
a  softer  paper  is  desirable.  Avoid  rolling  the  paper.  Sizes  in  common  use  are  18"  x  24",  12"  x  18" 
and  9"  x  12";  also  15"  x  22",  and  11"  x  15". 

Portfolio  or  Binding  Cover,  to  hold  paper  and  drawings. 

Notebook  Cover,  with  loose  sheets,  about  6"  x  9". 

Cloths,  a  white  cotton  cloth  about  14"  x  20",  hemmed,  to  place  materials  upon;  a  piece  of  cloth 
or  wash  leather  for  wiping  instruments;  and  a  small  dusting  cloth. 

Box,  to  hold  the  pencils,  erasers,  cloths,  and  other  small  articles. 

5.  Care  and  Arrangement  of  Equipment.     Next  in  importance  to  having 
good  instruments  and   materials  is    the    necessity   of   handling   them   properly, 
keeping  them  clean,  in  good  working  condition,  and  in  convenient,  orderly  arrange- 
ment. 


14  ESSENTIALS  OF  MECHANICAL  DRAFTINC 

Keep  the  fingers  clean,  and  the  table  and  materials  free  from  dust.  The 
T-square  and  triangles  especially  will  need  frequent  wiping.  A  paper  or  cloth  may 
be  fastened  over  a  part  of  the  drawing  to  protect  it  while  working  on  other  parts. 

Keep  instrument  case,  box,  and  locker  closed;  and  the  ink  bottle  in  the  stand, 
with  stopper  in  place. 

The  instruments  must  always  be  carefully  wiped,  and  properly  replaced  in 
the  case  when  through  working. 

Never  permit  ink  to  dry  in  the  pens,  or  upon  any  part  of  the  equipment. 

Take  every  precaution  to  insure  against  injury  to  the  points,  shanks,  etc.,  of  the 
instruments;  and  the  edges  of  the  scale,  board,  T-square,  triangles,  and  curve 
rulers. 

Adjustments  of  joint-screws  in  compasses  and  dividers  should  be  made  by 
the  instructor,  unless  otherwise  directed. 

Inaccuracy,  injury,  or  loss  of  any  part  of  the  equipment  should  be  reported 
immediately. 

The  table  should  be  so  placed  that  the  light  comes  from  the  left*  and,  if 
possible,  adjusted  to  such  a  height  that  the  student  may  stand  while  at  work. 

6.  Drawing  Board.     The  drawing  board  provides  a  flat  surface  upon  which 
to  secure  the  paper,  and  a  st.  edge  against  which  to  guide  the  head  of  the  T-square. 

Either  short  edge  may  be  selected  for  this 
purpose,  but  the  board  must  be  placed  so 
that  this  edge  will  be  at  the  left*,  and  no 
other  used  as  a  guiding  edge  in  elemen- 
tary work.  See  Fig.  48. 

For  convenience  in  working  and  to 
insure  firmness  and  freedom  in  the  use  of 
T-square  and  triangles,  the  paper  should 
be  placed  about  3"  from  the  left*  and 
lower  edges  of  the  board. 

Square  the  paper  with  the  board  by 
FIG  4-8  lining  up  one  of  its  edges  against  the  ruling 

edge  of  the  T-square.     Art.  7. 

Insert  tacks  about  T\"  from  each  corner,  pressing  the  paper  as  flat  as  possible 
with  the  hand;  then  with  thumb  or  tack-driver  force  the  heads  flush  with  the 
paper,  so  that  they  will  not  interfere  with  the  use  of  the  T-square,  etc.  Always 
remove  T-square  from  board  when  using  tack-driver,  and  avoid  marring  the 
board. 

After  paper  is  fastened,  the  board  may  be  inclined  by  means  of  a  book  or 
block  under  its  farther  edge, so  that  all  parts  of  the  drawing  may  be  more  nearly 
at  the  same  distance  from  the  eyes. 

Keep  paper  secured  to  the  board  until  drawing  is  completed.  If  removed 
before,  it  should  be  refitted  by  inserting  a  tack  in  one  of  its  upper  corners  and 
lining  up  the  most  important  horizontal  of  the  drawing  against  the  T-square. 

7.  T-square.      The  T-square  is  used  with  its  head  against  the  left  edge  of 
the  board.     Fig.  48.     The  upper  edge  of  the  blade  in  this  position  is  the  ruling 

*In  this  and  similar  working  directions,  left-handed  students  may  read  "right"  in  place  of  "left." 


INSTRUMENTS,  MATERIALS,  AND  THEIR  USE 


15 


FIG  49 


edge  for  all  hor.  lines,  and  guide  for  the  triangles 
when  drawing  lines  at  certain  Z  s.  Never  use  the 
lower  edge  of  the  blade  as  this  would  lead  to  errors 
difficult  to  trace.  As  the  Z  of  the  head  and  blade 
in  different  T-squares  is  apt  to  vary,  the  same  T-- 
square should  be  used  until  the  drawing  is  com- 
pleted. 

(a)       To  DRAW  A  HORIZONTAL  THROUGH  A  GIVEN 

POINT.      Slide  the  head  along  the  guiding  edge  of 

board  until  the  ruling  edge  passes  through  the  given 

pt.     For  preparation  and  use  of  pencil,  see  Art.  9.     Move  T-square  by  the  head 

only,  and  while  drawing  the  line  keep  head  and  blade  securely  in  position  by 

sliding  the  fingers  of  the  left  hand  along  the  blade  and  pressing  towards  the  right. 

Fig.  48. 

8.  Triangles.  The  triangles  (Fig.  49)  are  used  as  rulers  for  lines  at  Zs 
with  the  hor.  direction.  The  ruling  edges  of  the  45°  triangle  form  an  Z  of  90° 
and  two  of  45°  each.  Those  of  the  30°  .Y  60°  triangle  form  an  Z  of  90°,  one  of 
30°,  and  one  of  60°. 

(a)  TO    DRAW    A    LINE    AT    AN 
ANGLE    OF    30°,    60°,    45°,     OR    90°. 

Place  one  edge  of  the  correspond- 
ing 30°,  60°,  45°,  or  90°  Z  of  the 
triangle  against  the  T-square  and 
guide  the  pencil  along  the  other 
edge  of  the  Z .  Hold  T-square 
and  triangle  firmly  in  position  with 
left  hand.  Fig.  50.  When  the 
line  to  be  drawn  is  longer  than 

the  edge,  slide  the  T-square  until  ^^-Ju —  Fl  G  50 

the  required  length  is  obtained. 

Avoid  ruling  near  the  corners,  as  they  are  apt  to  be  rounded;  thus,  in  drawing 
a  line,  say  at  90°  with  a  given  hor.,  place  the  T-square  a  little  below  the  hor.  as 
shown.  Never  rule  against  the  inner  edges. 

(b)  TO  DRAW  A  LINE  AT  AN  ANGLE    OF   75°  OR 

15°.  Combine  the  triangles  and  T-square  as  in 
Fig.  51.  The  30°  Z  added  to  one  of  45°  gives 
an  Z  of  75°.  By  reversing  the  upper  triangle  as 
shown  by  dotted  lines,  one  of  its  edges  will  be  at 
15°.  By  placing  the  60°  angle  against  one  of  45°,  a 
line  at  75°  or  15°  in  the  opposite  direction  may  be 
obtained. 

(c)  TO     DRAW     A     PARALLEL     TO     ANY     GIVEN 

LINE.*     Combine  the  triangles  with  an  edge  of  one 


(see  dotted  triangle,  Fig.  52)  to  coincide  with  the 


FIG    51 


*When  the  given  line  is  hor.,  or  at  30°,  60°,  45°,  90°,  75°,  or  15°  with  the  hor.  direction,  the 
T-square,  or  triangle  and  T-square  combinations,  would  ordinarily  be  used. 


16 


KSSKNTIAI.S  UK   MI-XMI AMCAI. 


given  line  A-B;  then,  holding  the  second  securely, 
slide  the  first  until  the  edge  which  originally  co- 
incided with  A-B  is  in  the  required  position  C-D. 
(d)     To  DRAW  A  LINE  AT  30°,  60°,  45°,  90°,  75°, 

OR  15°  WITH  ANY  GIVEN   LINE.* 

For  80°,  60°,  45°,  or  90°,  Fig.  53.  Combine  the 
triangles  with  an  edge  of  one  ||  to  the  given  line 
A-B,  as  in  (c) .  Then  holding  this  triangle  securely, 
shift  the  second,  placing  it  against  the  ||  edge  of 
the  first,  so  that  one  of  its  edges  makes  the  re- 
quired Z  with  A-B. 

For  75°  or  15°,  Fig.  54.  It  is  evident  that  after 
the  second  triangle  has  been  placed  against  the  || 
edge  of  the  first,  the  first  in  turn  must  be  shifted  to 
give  the  required  Z . 

9.  Pencils  and  Writing  Pens.  The  hard 
pencil  is  used  for  ruling  lines  against  the  T-square, 
triangles,  and  curve  rulers;  the  medium  pencil  for 
all  freehand  penciling,  lettering,  etc.,  and  the  pens 
for  inking  freehand  lines,  lettering,  etc. 

(a)  To  SHARPEN  THE  PENCILS.  Sharpen  the 
end  not  bearing  the  grade  stamp.  Hold  the  inner 
side  of  both  wrists  firmly  against  the  body,  with 
the  knife  blade  nearly  flat  against  the  upper  side 
of  the  pencil  and  its  cutting  edge  to  the  right. 
Cut  with  an  outward  wrist  movement,  removing 
the  wood  in  long  thin  shavings,  tapering  it  evenly 
down  to  but  not  cutting  the  lead,  and  so  that  about 
|"  of  the  latter  is  exposed.  Fig.  55.  This  method 

gives  greater  control  of  the  knife  and  lessens  the  liability  of  soiling  the  fingers.  The 
lead  should  be  pointed  by  means  of  a  lead-pointer.  Hold  pointer  in  the  left  hand, 
away  from  the  table,  and  the  pencil  so  that  the  entire  length  of  lead  will  be  tapered. 
Taper  the  lead  of  the  freehand  pencil  to  a  fairly  sharp  pt.  by  rubbing  it  back 
and  forth,  at  the  same  time  rolling  the  pencil  between  thumb  and  fingers.  Fig.  (a). 

The  lead  of  the  ruling  pencil  should  be  tapered 
to  a  sharp  conic  pt.,  or  to  a  wedge  pt.,  as  the  in- 
structor directs.  The  wedge  pt.  (Fig.  b)  retains  its 
sharpness  longer  and  fits  more  closely  against  the 
ruling  edge.  It  is  obtained  by  first  tapering  the 
lead  slightly  as  for  the  freehand  pt.,  and  then 
rubbing  the  lead  upon  opposite  sides  to  form  a 
short,  sharp  edge  at  the  end. 

Rub  the  leads  frequently  to  keep  them  in 
proper  condition. 

*Whcn  the  given  line  is  hor.,  or  at  30°,  60°,  45°,  90°,  75°,  or  15°  with  the  hor.  direction,  the 
T-square,  or  triangle  and  T-square  combinations,  would  ordinarily  be  used. 


FIG  54 


FIG.  55 


INSTRUMENTS,  MATERIALS,  AND  THEIR  USE 


17 


FIG  56 


(b)  USE  OF  THE  FREEHAND  PENCIL.     Hold 
pencil  lightly,  about    1^  inches  from  the  point. 
The  relation  of  the  pencil  to  the  line,   and  the 
direction  of  the  stroke,  should  usually  be  as   in- 
dicated in  Fig.  56.     Face  the  paper  squarely  and 
avoid  turning  it  while  drawing. 

All  lines  should  be  drawn  with  as  few  strokes 
as  possible. 

Draw  lightly  at  first  and  correct  any  portion 
by  drawing  a  second  line  before  erasing.  Finally 
strengthen  the  line  to  make  it  clear  and  even. 
Practice  in  arm  and  finger  movements  before 
drawing  will  aid  in  acquiring  necessary  freedom. 

(c)  USE  OF  THE  RULING  PENCIL.     Hold  pen- 
cil as  nearly  upright  as  possible,  with  flat  side  of 
lead  against  the  ruling  edge.     Steady  pencil  in 
this  position  by   resting  the  tip  of  the  third  or 
fourth  finger  upon  the  ruler.     Fig.  48.      This  per- 
mits the  edge  of  the  lead  to  wear  evenly  and  give 
uniform  lines.     Bear  lightly, — much  pressure  will 
dull  the  lead  too  rapidly,  make  uneven  lines,  or 
form   depressions  which  cannot  be  erased.     The 
result  should  be  very  fine,  clear  lines. 

The  pencil  should  always  be  moved  from  left  to  right,  the  student  turning  his 
body  or  the  board  when  necessary,  that  he  may  face  the  ruling  edge  squarely 
and  do  this  more  readily.  Thus  in  drawing  verts,  face  to  the  left,  and  draw 
away  from  the  T-square.  (See  Fig.  50.)  If  ruled  against  a  right-hand  edge,  the 
pencil  is  apt  to  glide  away  from  the  ruler  and  cause  a  break  in  the  line.  Watch 
the  point  constantly  as  it  is  moved  along.  Never  rule  a  line  in  a  shadow,  and 
never  rule  backward  over  a  line. 

In  drawing  parallels  move  the  rmer  from  one  position  to  the  other  in  such 
manner  that  the  preceding  line  and  space  will  not  be  covered.  Thus,  in  drawing 
II  hors.,  move  T-square  downwards;  in  ||  verts.,  move  triangle  to  the  right. 

(d)  USE  OF  WRITING  PENS.      Handle  the  pen  the  same  as  freehand  pencil. 
Use  little  ink  and  aim  to  secure  uniform, 

even  lines.   Exercise  special  care  in  work- 
ing over  lines  to  avoid  inj  uring  the  paper. 

10.  Needle-point.  The  needle-point 

is  used  to  set  off  distances  from  the  scale,     \ 
and  to  fix  pts.  of  line  intersections  which 
might  otherwise  be  erased  or  lost.    Hold 
needle  upright  and  make  the  smallest 
puncture  that  can  be  seen. 

11.  Scales.     The  scale  is  used  for 

setting  off  measurements.    It  must  never  (p)  FIG.  57 


(a) 


AW'Vm 

AZ 

'^g\\1V\^ 


v 


iv 


KSSKNTI.M.S  <>l     MKCII AN1CAI.   DUAITI.Vi 


be  used  as  a  ruler  for  drawing  lines.     The  scales  shown  in  Fig.  57  are  graduated 
as  described  in  Art.  4. 

(a)  When  the  drawing  is  made  so  that  each  inch  or  fraction  of  an  inch  of 
measurement  upon  it  is  equal  to  the  corresponding  measurement  on  the  object 
itself,  the  drawing  is  said  to  be  full  size  or  to  a  scale  of  12"  to  afoot. 


FIG.   58 

When  for  convenience  or  necessity  the  drawing  is  made  smaller  or  larger 
than  full  size  each  unit  of  measurement  is  made  smaller  or  larger  in  proportion, — 
thus  when  drawn,  say  half  size  or  6"  to  afoot,  each  half  inch  on  the  drawing  repre- 
sents one  inch  on  the  object;  each  quarter  inch  a  half,  and  so  on. 

Full  size  measurements  are  obtained  from  a  scale  graduated  to  full  size  inches 
and  fractions. 

Half  size  measurements  are  usually  obtained  from  a  full  size  scale  by 
simply  reading  each  half  inch  on  the  scale  as  one  inch,  each  quarter  inch  as  a 
half,  etc.  Quarter,  eighth,  and  sixteenth  size  can  be  obtained  in  like  manner, 
or  from  scales  graduated  to  3",  1£",  and  \"  to  1 
ft.  respectively. 

In  scales  graduated  to  feet  and  inches  the  first  unit, 
when  large  enough,  is  divided  to  represent  inches  and 
fractions.  For  example,  in  the  scale  of  \\"  to  1  ft. 
the  first  unit  is  divided  into  12  eighth-inch  parts  to 
represent  inches,  and  each  of  these  subdivided  to  repre- 
sent halves  and  4ths.  Fig.  57  (b). 

When  lOths,  20ths,  30ths,  etc.,  of  an  inch  are  required, 
an  engineer's  or  decimal  scale  is  used.  Any  desired  scale 
may  be  drawn  by  division  of  B,  line  into  the  required  pro- 
portional parts.  See  Art.  30. 

For  method  of  determining  the  size  or  scale  to  be  used, 
see  Art.  100 (c). 

(b)       To     SET     OFF     A     MEASUREMENT     (as     Say      2^")- 

Apply  the  scale  directly  to  the  line,  with  the  "0"  division 
exactly  at  the  end  of  the  line,  and  the  needle-point  at  the 
2i"  division.  To  set  off,  say  2  ft.  and  2$",  place  the  di- 
vision representing  2j"  at  one  end  of  the  line  and  a  pt.  at 
that  representing  2  ft. 

Never  transfer  measurements  from  the  scale  with 
compasses  or  dividers.  Successive  measurements  on  a 


\ 


FIG.  59 


INSTRUMENTS,  MATERIALS,  AND  THEIR  USE  19 

line  should,  so  far  as  possible,  be  set  off  without  shifting  the  scale,  so  that  an 
error  in  one  distance  may  not  affect  all. 

12.  Protractor.     A  Protractor   (Fig.   58)   is  a  scale  used  in  laying  off  and 
measuring  Z  s.     Its  measuring  edge  is  graduated  to  degrees  and  fractions  (usually 
to  half  degrees),  and  the  degrees  numbered  to  read  from  0  to  180°  in  both  direc- 
tions. 

(a)       To    DRAW    A    LINE    AT   ANY    GIVEN    ANGLE    WITH    A    GIVEN    LINE    (say    40°). 

Place  the  protractor  with  its  180°  line  0-0  against  the  given  line  A-B  and  center 
C  at  the  given  vertex  A.  Now  place  a  pt.,  D,  at  the  40°  mark.  Remove  the 
protractor  and  draw  A-D,  the  required  line. 

13.  Compasses.     The   compass  set   (Fig.   59)   is  used  for  drawing  circular 
curves.     The  needle  should  first  be  adjusted  as  follows: — Release  clamp  screw 
A,  and  remove  the  lead  holder.     Insert  the  pen  in  place  of  the  latter,  and  clamp 
securely.     Then  set  the  needle  so  that  the  point  of  its  shouldered  end  will  be  even 
with  the  pen  point.     Once  set,  the  needle  should  not  be  changed.     The  lead 
only  will  need  resetting,  as  it  wears  away. 

(a)  TO    PREPARE    THE    COMPASSES    FOR    PENCILING. 

Place  a  4H  compass  lead,  about  1"  long,  in  the  holder, 
and  taper  it  as  directed  for  the  ruling  pencil.  Refit 
the  holder  and  adjust  the  lead  to  the  length  of  the 
needle,  with  its  edge  so  placed  that  a  fine,  even  line 
will  result  when  the  compasses  are  revolved. 

(b)  To  DRAW  A  CIRCLE.      Open  the  compasses  and 
adjust  the  legs  at  the  joints   so   that  both   lead  and 
needle  will  be  at  right  Z  s  to  the  paper  while  drawing. 
This  is  necessary  to  prevent  the  lead  from  wearing  un- 
evenly and  the  needle  from  digging  into   the   paper. 

The  puncture  should  be  barely  visible.  riG.oO 

Hold  the  compasses  by  the  handle  only,  with  thumb  and  first  two  fingers 

(see  Fig.  60)   and  always  revolve  it  around  to  the  right  (clockwise).     Bearing 

lightly  and  evenly  upon  the  lead  point,  draw  the  curve  with  one  continuous 

motion,  stopping  exactly  at  the  end  of  the  revolution  to  avoid  widening  the  line. 
In  drawing  a  O  or  arc  of  given  rad.,  first  set  off  the  rad.  upon  a  C.L.,  and,  in 

placing  the  compass  point  at  the  center,  steady  the  needle  with  a  finger  of  the 

left  hand. 

Changes  of  rad.  should  be  made,  so  far  as  possible,  with  the  right  hand  only, 

and  care  taken  not  to  enlarge  the  center. 

(c)  To   INK   CIRCLES  AND   CIRCULAR   ARCS.     Insert  the   pen   and   clamp  it 
securely.      Clean,  fill,  and  set  the  pen,  as  directed  in  Art.  17 (a).      Open  compasses 
to  the  rad.  of  the  penciled  curve  and  adjust  the  legs,  as  directed  in  (b).     To  give 
clean-cut  lines,  both  blades  must  bear  evenly  upon  the  paper.     The  directions 
for  penciling  apply  also  to  inking.     Before  placing  the  pen  point  upon  the  curve, 
the  compasses  should  first  be  revolved  over  the  line,  in  space,  to  make  sure  that 
the  inked  line  will  pass  exactly  through  the  desired  pts.     Errors  caused  by  enlarge- 
ment of  centers  may  thus  be  avoided.     Do  not  go  over  a  line  a  second  time. 


90 


1 .»!  .NTI.M.S  ol     MI.CIIANICAI.   DUA1-TINC! 


FIG.  61 


FIG   62 


(d)  LENGTHENING  BAR.  When  the  rad.  is  too  great  to  admit  of  placing  tin- 
points  _L  to  the  paper,  the  lead  or  pen  log  should  be  extended  by  means  of  tin- 
lengthening  bar.  In  this  case,  the  needle  leg  may  be  steadied  with  the  left  ha  ml 
and  the  drawing  point  moved  with  the  other,  care  being  taken  not  to  change  the  rad. 

14.  Bow  Compasses.  The  bow  pencil  (Fig. 
r>l)  and  bow  pen  (Fig.  62)  are  used  for  drawing 
small  Os  and  arcs  for  which  the  large  com- 
passes are  not  convenient.  Do  not  use  them 
for  radii  over  f  ".  The  directions  for  prepar- 
ing and  using  the  bow  compasses  are  much 
the  same  as  for  the  large  compasses.  The 
needle  must  project  slightly  beyond  the  pen  or 
lead  point,  and  the  lead  be  tapered  more  nearly 
to  a  pt.  To  enable  the  points  to  be  brought 
close  together,  the  needle  is  generally  flattened 
on  its  inner  side.  The  rad.  is  adjusted  by 
turning  the  thumb-nut  on  the  connecting  bar. 

Before  turning,  spring  the  points  together  so 
that  the  wear  of  the  screw  thread  may  be 
lessened  and  the  adjustment  made  more  readily. 

15.  Dividers.  The  dividers  or  spacers  (Fig.  63)  are  used  for  transferring 
measurements  from  one  part  of  a  drawing  to  another,  and  for  setting  off  equal 
distances  on  a  line  when  they  cannot  readily  be  laid  off  by  means  of  the  scale. 
Handle  the  dividers  in  the  same  general  way  as  the  compasses. 

In  some  dividers  one  leg  is  furnished  with  a  hairspring  and  nut  by 
means  of  which  this  leg  may  be  moved  for  slight  changes  of  adjustment. 

(a)       TO     DIVIDE     A    STRAIGHT    LINE     OR    A     CIRCULAR     CURVE    INTO 

ANY  NUMBER  OF  EQUAL  PARTS  (say  3).  Open  the  dividers  to  a  dis- 
tance equal  (by  eye)  toiof  the  line  to  be  divided,  place  one  of  its  points 
upon  the  end  of  the  line  and  revolve  the  dividers  until  the  other  point 
is  exactly  on  the  line.  Proceed  in  this  manner,  revolving  alternately 
in  opposite  directions,  until  the  distance  taken  has  been  set  off  three 
times.  Never  remove  both  points  at  the  same  time.  If  the  -distance 
taken  does  not  apply  exactly  it  must  be  increased  or  diminished  by 
an  amount  equal  to  i  of  the  difference,  and  the  trial  repeated  until  the 
line  is  equally  divided.  No  pts.  should  be  made  visible  until  the 
divisions  have  been  verified  as  correct. 

16.  Bow  Dividers.     Fig.  64.      These  are  used  for  small  distances, 
and  in  the  same  general  manner  as  the  large  dividers.    The  points  are 
adjusted  as  directed  for  bow  compasses,  Art.  14. 

17.  Ruling  Pen.     The  ruling  pen  (Fig.  65)  is  used  for  inking  all 
lines  other  than  circular  curves. 

(a)     To  FILL  AND  CLEAN  THE  PEN.      Before  filling  the  pen,  moisten 
a  folded  end  of  the  penwiper  and  draw  it  gently  between  the  blades. 
FIG  63      When  clean  and  dry  bring  the  blades  together  at  the  point  by  means 


INSTRUMENTS,  MATERIALS,  AND  THEIR  USE 


21 


FIG.  65 


of  the  thumb-screw;  then  holding  the  pen  upright/ 
not  over  the  drawing,  insert  the  ink  between  the 
blades  with  the  filler.  Do  not  fill  above  \"  from 
the  point,  otherwise  the  ink  will  flow  out  too  freely. 
See  also  that  there  is  no  ink  on  the  outside  of  the 
point,  as  this  will  widen  the  line,  make  it  ragged,  or 
cause  a  blot.  Replace  the  stopper  immediately  to 
prevent  the  ink  from  thickening. 

Having  filled  the  pen,  set  the  blades  to  the  re- 
quired width  of  line.     Always  try  the  pen  on  a  piece 
of  waste  paper  before  using  it  on  the  drawing.     To 
insure  its  flowing  freely,  the  amount  of  ink  in  the  pen 
must  be  kept  as  nearly  as  possible  the  same.     Avoid 
having  to -piece  out  a  line.     As  the  ink  dries  rapidly, 
the  pen  must  be  cleaned  and  refilled  frequently.     In 
doing  this,  it  is  not  necessary  to  open  the   blades. 
The  setting  should  remain  unchanged  until  all  lines  of  the  same 
width  are  inked.     If  the  pen  fails  to  work,  it  should  be  sharpened 
by  an  experienced  person.     Never  put  the  pen  aside  without  care- 
fully cleaning  it. 

(b)  To  RULE  A  LINE.  Hold  and  steady  the  pen  as  directed 
for  the  pencil  (Art.  9  (c)),  the  first  finger  resting  on  the  flat  side  of 
the  pen  above  the  thumb-screw  and  the  second  against  the  edges  of 
the  blades  as  shown  in  Fig.  50. 

Adjustments  for  width  are  made  with  thumb  and  second  finger  of  the  same  hand. 

Guard  against  getting  the  pen  point  too  close  to  the  ruling  edge  by  placing 
the  latter  slightly  away  from  the  line,  as  shown.     A  slight  downward  pressure 
only  should  be  necessary,  but  the  points  of  both  blades  must  bear  evenly  upon 
the  paper  to  give  a  clean-cut  line.     Bear  lightly  against  the  ruling 
edge  to  prevent  varying  the  width  of  the  line.     Always  steady  the 
hand,  and  move  the  pen  from  left  to  right  as  directed  for  penciling. 
Just  before  reaching  the  end  of  the  line  stop  the  arm  movement 
and  complete  the  line  with  a  finger  movement,  then  lift  the  pen 
immediately  and  move  the  ruling  edge  away  from  the  line. 

In  ruling  curves,  turn  the  pen  gradually  so  that  the  blades  will 
not  be  at  an  Z  with  the  ruling  edge.  Art.  18.  Never  use  the  ruling 
pen  freehand. 

18.  Curve  Rulers.  These  are  used  for  ruling  curves  that 
cannot  be  drawn  with  the  compasses.  They  are  made  in  various 
shapes  and  sizes.  Two  of  the  most  serviceable  are  shown  in 
Figs.  66,  67. 

(a)  To  RULE  A  CURVE,  ABC.  Fig.  67.  First  sketch  the 
curve  lightly  with  freehand  pencil,  through  previously  determined 
pts.  Now  find,  by  trial,  a  portion  of  the  ruler  which  will  fit  as 
much  of  the  curve  as  can  be  ruled  conveniently  at  one  time,  as  A-B, 
and  true  up  that  part  by  tracing  over  it  with  the  ruling  pencil. 


FIG.  66 


22  ESSENTIALS  OF  MECHANICAL  DRAFTING 

Match  succeeding  parts  in  the  same  manner,  making  the  edge  fit  over  a  portion 
of  each  preceding  part  to  insure  an  even,  unbroken  line.  The  freehand  penciling 
should  not  be  omitted,  as  the  tendency  would  be  to  make  the  ruled  line  curve  out 
too  much  or  too  little.  Having  trued  the  line,  rub  the  soft  rubber  lightly  over  it. 

In  inking  the  curve,  use  the  ruling  pen  as  described  in  Art.  17(b).  In  ruling 
curves  symmetrical  about  one  or  more  axes,  as  ellipses,  helices,  etc.,  the  portion 
of  the  ruler  used  for  one  part  should  be  noted  and  used  for  corresponding  parts. 
Sharp  turns  at  ends  of  axes  should  first  be  drawn  by  means  of  compasses,  the 
centers  being  taken  in  the  axes  and  care  taken  to  use  the  proper  rad.  and  length 
of  arc. 

(b)  Non-circular  curves  may  often  be  approximated  throughout  by  tangent 
arcs.  Thus,  in  inking  the  curve  shown  by  the  fine  line  in  Fig.  68,  beginning  say 
at  A,  determine  by  trial  the  center  and  rad.  of  as  much  of  an  arc  as  will  practically 
coincide  with  the  curve.  Ink  this  arc;  then,  changing  the  center  and  rad.,  ink 
the  next  portion;  note  that  the  centers  must  be  on  the  line  through  the  pt.  of 
tangency. 


FIG.  68 


19.  Erasers.  (a)  The  soft  rubber  is  used  for  pencil  erasing  and  paper 
cleaning.  Keep  paper  as  clean  as  possible  from  the  start.  (See  Art.  5.)  The 
softest  rubber  is  liable  to  roughen  the  paper,  making  it  difficult  to  keep  clean 
and  to  obtain  sharp  lines.  In  removing  a  line,  rub  lengthwise.  Avoid  much 
pressure  and  always  remove  dust  before  proceeding  with  drawing.  When  a 
drawing  is  finished  in  ink,  the  eraser  may,  if  necessary,  be  passed  lightly  over  the 
entire  surface,  care  being  taken  to  avoid  dulling  the  lines. 

(b)  The  hard  rubber  is  used  for  ink  erasing.  The  part  to  be  removed  should 
first  be  allowed  to  dry.  Care  must  be  taken  not  to  injure  the  surface.  Never 
use  a  knife.  If  the  surface  is  roughened  by  erasing,  smooth  it  as  well  as  possible 
with  the  finger  nail.  A  thin  card  or  piece  of  celluloid  with  narrow  openings  can 
be  used  to  protect  adjacent  parts  when  erasing. 


CHAPTER  III 
PENCILING    AND    FINISH    RENDERING 

20.  Layout  of  the  Sheet,  (a)  MARGIN  LINES.  To  improve  the  general 
appearance  of  the  drawing  and  to  insure  keeping  all  lines  and  figures  a  safe  distance 
from  the  edges  of  the  sheet,  it  is  customary  to  rule  border  lines  with  uniform 
marginal  spaces  at  top,  bottom,  and  sides.  See  Fig.  69. 

Assuming  the  paper  to  be  H"xl5",  the  required  border  10"  x  13|",  and 
marginal  spaces  |",  proceed  to  lay  out  the  border  and  trim  lines  as  follows: 
Having  tacked  the  paper  to  the  board  (Art.  6)  set  off  two  pts.  near  the  left  edge, 
\"  and  10£"  respectively,  above  the  lower  edge,  Art.  11.  Through  these  pts. 
draw,  light  horizontals  across  the  sheet,  Art.  7.  On  the  lower  hor.,  set  off  a  pt. 
\"  from  the  left  edge;  \"  from  this  place  a  second;  13£"  from  the  second,  a 
third;  and  \"  from  that  a  fourth.  Through  these  pts.,  draw  the  90°  lines 
(verticals),  Art.  8.  Test  distances  between  verts,  at  the  top  and  hors.  at  the 
right  with  the  scale. 


* 


IQH5Z 


•SHfcM     I    -'OB 


4 


J    5M1TM     17-A 


FIG. 69 


When  the  sheet  is  completed  and  removed  from  the  board,  the  \"  strips  con- 
taining the  tack  holes  are  to  be  cut  off,  thus  leaving  the  sheet  11"  x  14|". 

(b)  LOCATION  OF  NAME,  TITLE,  ETC.  On  elementary  drawings  these  may 
be  lettered  as  shown.  For  titles,  etc.,  on  working  drawings,  see  Art.  102.  For 
instructions  in  lettering,  see  Art.  25. 

21.  Constructive  Stage  of  the  Drawing.  All  lines  should  be  penciled  first 
in  uniform,  fine  full  lines,  as  indicated  in  Fig.  209(a),  (b),  (c).  Then,  to  avoid 
errors  in  the  finishing  stage  (Art.  22), lines  representing  edges  and  outlines  of  the 
object  should  be  gone  over  with  a  slightly  firmer  pressure,  and  the  hidden  parts 


24  ESSENTIALS  OF  MECHANICAL  DRAFTING 

dashed  as  in  Fig.(d).  The  general  order  of  penciling  different  kinds  of  drawings 
is  indicated  in  Art.  3. 

To  avoid  the  necessity  of  piecing  out,  make  the  lines  first  of  indefinite  length. 
Be  sure  that  all  measurements  are  set  off  upon  definite  lines  and  that  lines  intended 
to  pass  through  particular  pts.  actually  do  so.  No  part  of  the  penciling  should 
he  slighted.  Inaccuracies  can  seldom  be  corrected  in  the  process  of  finishing. 
Aim  to  do  no  erasing  until  the  penciling  is  completed. 

To  secure  greater  accuracy  and  economy  of  time,  similar  operations  should 
be  grouped.  Thus,  draw  all  lines  that  can  be  ruled  with  the  T-square  and  triangles 
in  one  position  at  the  same  time,  and  ||s  of  one  set  before  commencing  those  of 
another.  When  using  the  scale  or  dividers,  set  off  all  distances  possible  at  once. 
When  using  the  compasses  work  the  circular  curves  in  the  same  way,  drawing 
those  having  the  same  rad.  at  one  setting  of  the  instrument,  etc.  As  st.  lines 
can  be  drawn  tangent  to  curves  more  accurately  than  the  reverse,  pencil  the  curves 
first  whenever  possible. 

To  enable  the  method  of  procedure  to  be  readily  followed,  all  lines  used  in 
making  the  drawing  should  be  left  upon  the  sheet  until  it  has  been  approved. 

22.  Finishing  Stage  of  the  Drawing.  For  greater  distinctness  and  per- 
manence the  drawing  is  usually  lined-in  or  finished  with  ink,  either  by  going  over 
the  lines  on  the  original  or  by  making  a  tracing  as  in  Art.  106.  Drawings  not 
intended  for  continued  use  or  of  which  no  copies  are  needed  are  often  finished  in 
pencil. 

Do  not  commence  the  finishing  stage  until  the  constructive  stage  is  completed 
and  approved.  See  that  the  drawing  surface  is  free  from  dust. 

(a)  INKING.     Be  careful  to  make  sharp,  even  lines,  and  see  that  all  lines 
begin  and  end  exactly  where  it  is  intended  that  they  should.     To  prevent  lines 
running  together  at  their  intersection,  see  that  the  first  is  thoroughly  dry  before 
inking  the  second.     Do  not  use  a  blotter. 

(b)  FINISHING  IN  PENCIL.     If  the  drawing  is  to  be  finished  in  pencil,  the  aim 
should  be  to  secure  as  nearly  as  possible  the  accuracy  and  distinctness  of  an  inked 
drawing.     The  medium  pencil  should  be  used,  at  least  for  strengthening  object 
lines.     In  finishing  dashed  lines  it  is  not  necessary  to  erase  lines  of  the  constructive 
stage  between  dashes  as  they  will  not  be  prominent  if  the  finish  lines  are  properly 
emphasized. 

(c)  LINE  CONVENTIONS.     The  different  purposes  of  the  lines  of  the  drawing 
are  indicated  by  varying  their  character,  width,  or  color.     The  conventions  shown 
in  Fig.  70  are  commonly  used  on  drawings  finished  wholly  in  black  ink  or  in 
pencil,  for  the  purposes  stated  below.     They  are  suitable  in  width,  length  of 
dash,  etc.,  for  ordinary  drawings. 

A — Visible  lines  of  objects  in  all  required  views*  and  edges  in  developments. 

B — Hidden  lines  of  objects  in  all  required  views.  As  a  rule  end  dashes  should 
touch  the  limiting  lines. 

C — Visible  lines  of  objects  in  auxiliary  views  used  in  determining  required  views. 
Whenaux.  views  show  visible  lines  only,  they  may  be  finished  as  construction  lines. 

*When  shadow  or  shade  lines  are  used,  shade  the  curved  edges  as  each  is  lined-in.  (See  Art. 
23(c).)  St.  shadow  lines  are  left  in  pencil  until  all  but  the  border  is  finished. 


PENCILING  AND  FINISH  RENDERING  25 

D— Hidden  lines  of  objects  in  auxiliary  views  used  in  determining  required  views. 

E — Traces  of  projection,  section,  and  base  planes;  center  lines,  and  axes.  Dashes 
of  section  traces,  center  lines  and  axes  should  extend  about  f "  beyond  the  view 
or  part  on  which  they  are  drawn. 

F — Construction  (working)  lines  required  to  show  method  of  construction;  pro- 
jectors, and  extension  lines.  Dashes  of  extension  lines  should  not  touch  lines 
of  views,  and  should  extend  about  rV"  beyond  arrowheads  of  dimension  lines. 

G — Dimension  lines,  and  pointers. 

H — Break  lines.     Same  as  edges,  etc.,  but  should  be  drawn  freehand. 

I — Arrowheads,  figures,  notes,  titles,  etc.  These  should  be  penciled  freehand, 
and  in  inked  drawings  always  finished  in  black.  See  Fig.  74. 

J — Section  lining.     Same  as  C,  or  as  in  Fig.  187. 

K — Line  shading.     As  in  Art.  24. 

L — Straight  shadow  and  shade  lines  of  all  required  views,  and  border  lines. 

Red  ink.  In  inked  drawings,  the  use  of  full  red  lines  for  all  lines  under  C,  E, 
F,  and  G,  and  dashed  red  for  those  under  D  is  more  economical  of  time  and  makes 
the  required  views  more  prominent. 

A WIDTH  2 

B       ,,       2 

C  >•        I 

D  ••        I 


FIG.  70 


(d)  ORDER  OF  FINISHING..  It  is  desirable  to  complete  all  lines  and  parts  of 
similar  character  before  proceeding  with  those  of  another,  in  the  general  order 
indicated  in  Art.(c),  in  observing  which,  similar  operations  should  be  grouped 
as  in  the  following:  — 

Circles  and  arcs,  beginning  with  the  smallest;  non-circular  curves;  horizontals, 
beginning  with  those  at  the  top;  verticals,  beginning  with  those  at  the  left;  obliques, 
beginning  with  those  obtainable  by  T-square  and  triangle  combinations. 

Do  not  use  a  triangle  alone,  unless  necessary. 

In  finishing  break  lines,  arrowheads,  figures,  and  notes,  etc.,  work  from  upper 
part  of  sheet  downward. 

23.  Shadow  Lining.  In  practical  drawings,  visible  edges  and  outlines  of 
objects  are  generally  indicated  by  full  lines  of  uniform  width.  It  is  sometimes 
desirable,  however,  to  finish  certain  lines  wider  than  the  others  for  the  purpose  of 
giving  an  appearance  of  relief  to  the  drawing  and  to  indicate  the  relative  positions 
of  the  surfaces  more  clearly. 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


x(a)     B      X     (b)  X      (c) 


CONVENTIONAL    SHADOW     LINES 
FIG.   71 


PENCILING  AND  FINISH  RENDERING  27 

.  (a)  If  a  cube  placed  as  in  Fig.  71  (a)  be  lighted  by  ||  rays  coming  over  the 
left  shoulder,  in  the  direction  of  diagonal  A-B,  it  is  evident  that  the  upper,  left, 
and  front  faces  will  be  in  the  light,  the  others  in  the  shade. 

Lines  separating  light  from  dark  surfaces  are  called  shade  lines.  The  visible 
shade  lines  of  the  cube  will,  therefore,  be  the  lower  and  right  lines  in  the  front  view, 
and  the  upper  and  right  lines  in  the  top  view,  and  may  be  indicated  by  broad 
lines  as  shown. 

Assuming  the  same  direction  for  the  light,  the  visible  shade  lines  in  all 
rectangular  objects  and  parts,  whose  axes  are  ||  or  _L  to  the  planes  of  the  views, 
would  have  the  same  general  locations  as  in  the  cube.  In  such  cases  the  shade 
lines  can  usually  be  determined  by  eye.  The  determination  of  the  actual  shade 
lines  in  all  cases,  however,  would  involve  considerable  time  and  labor,  and  their 
locations  would  frequently  be  such  as  to  complicate  the  drawing  rather  than 
aid  in  explaining  the  form  of  the  object;  hence  it  is  customary  to  disregard  the 
actual  shades  and  shadows  and  to  apply  the  broad  lines  in  such  manner  that 
they  will  indicate  edges  separating  visible  from  hidden  surfaces  only,  and  so  as 
to  produce  an  effect  of  narrow  shadows  cast  by  the  object. 

It  is  convenient  to  regard  these  lines  as  shadow  lines  rather  than  as  shade  lines. 

To  further  simplify  the  application,  most  draftsmen  shade  or  shadow-line 
all  views  the  same  as  the  front. 

The  aid  given  by  such  lines  in  reading  a  drawing  will  be  apparent  from  the 
figures. 

The  directions  for  this  conventional  method  may  be  stated  as  follows: — 

(b)  Shadow-line  the  lower  and  right  lines  of  intersection  between  visible  and 
hidden  surfaces  in  all  views,  regardless  of  the  actual  shades  or  shadows  of  the 
object.     This  includes  both  sharp  and  slightly  rounded  edges.     Lines  not  repre- 
senting such  edges  are  generally  shadow-lined  only  when  surfaces  extend  back 
from  them  at  right   Zs  to  the  plane  of  the  drawing,  in  order  to  indicate  more 
clearly  that  they  do  not  represent  edges.     A  sectional  view  is  generally  shadow- 
lined  just  as  if  the  portion  shown  were  complete.     Lines  representing  broken 
surfaces  are  preferably  not  shadow-lined.     Never  shadow-line  the  line  of  inter- 
section between  visible  surfaces,  nor  dashed  lines. 

(c)  To  SHADOW-LINE  A  DRAWING.     In  determining  the  shadow  lines  remember 
that  the  rays  of  light  will  be  at  45°  down  to  the  right  in  all  views  and  that  all 
views  are  shadow-lined  in  the  same  manner.     Limiting  rays  may  be  penciled  as 
shown.     Each  edge  to  be  shadow-lined  should  be  indicated  by  a  mark  upon  the 
line  before  inking  is  begun.     Drawings  are  never  shadow-lined  in  pencil. 

Straight  Lines.  A  st.  shadow  line  should  be  ruled  the  required  width  (3)  at 
one  stroke  of  the  ruling  pen,  and  its  width  added  on  the  outside  of  the  pen- 
cil line. 

Circular  Curves.  First  ink  the  curve  in  the  usual  manner  (width  2).  Then 
taking  a  second  center  below  and  to  the  right  of  the  first,  on  a  45°  line,  at  a  dis- 
tance equal  to  width  3  and  without  change  of  rad.  or  setting  of  pen,  draw  an  arc 
on  the  outside  or  inside  of  the  first  curve  according  to  the  required  location  of 
the  shadow  portion.  (See  Fig.  71  (h).)  If  an  uninked  portion  remains  between 
the  arcs,  spring  the  instrument  slightly  to  fill  it  in. 


28  ESSENTIALS  OF  MECHANICAL  DRAFTING 

Irregular  Curves.  First  ink  the  widest  portion;  then  with  the  ruling  pen  set 
to  width  2,  blend  this  carefully  into  the  rest  of  the  curve. 

24.  Line  Shading.  This  is  a  conventional  method  of  producing  an  effect 
of  light  and  shade,  corresponding  to  that  upon  the  object  itself  by  means  of  lines, 
usually  of  graded  widths  and  spacing.  Fig.  72.  It  is  used  on  drawings  intended 
for  illustrative  purposes  and  in  cases  where  it  is  necessary  to  indicate  the  direction 
of  certain  surfaces  more  clearly  than  would  be  possible  by  the  mere  outline  or  by 
shadow  lines. 


(d) 


FIG    72 


(a)  The  direction  of  the  light  is  assumed  as  for  shadow-lining  (Art.  23) ;  the 
same  for  all  views.  The  right  and  lower  sides  are,  therefore,  the  dark  sides  on 
convex  surfaces  and  the  left  and  upper  on  concave  surfaces.  In  regular  curved 
surfaces  the  darkest  portion  is  about  £  of  the  rad.  from  the  center.  It  may  be 
accurately  determined  as  shown  in  top  view  of  Fig. (a).  The  spacing  may  be 
indicated  in  pencil,  as  below  Fig. (a).  Sha3e  first  the  dark  portion,  beginning 
at  about  }  of  the  rad.  from  the  center.  Use  fine  lines  only  on  the  light  side, 
somewhat  farther  apart  than  on  the  dark,  and  stop  at  about  5  of  the  rad.  from 
the  center.  On  small  parts  shade  dark  side  only.  (See  Fig.(g).)  Figs,  (a)  and  (i) 


PENCILING  AND  FINISH  RENDERING  29 

illustrate  methods  of  shading  fillets,  beads,  etc.,  on  large  drawings;  Figs.(e), 
(f),  (g),  for  small  drawings.  The  shading  of  conic  surfaces  by  ||  lines,  as  shown, 
is  less  difficult  and  generally  more  satisfactory  than  by  radial  lines.  When 
desirable  to  shade  plane  surfaces,  the  method  in  Fig.(d)  may  be  used. 

25.  Lettering.  The  names,  titles,  notes,  and  dimensions  required  on  drawings 
should  be  carefully  lettered  in  even,  well-proportioned,  and  well-spaced  characters. 

The  styles  shown  in  Fig.  73  are  those  most  generally  used. 

The  inclined  Gothic  differs  from  the  vertical  only  in  the  slant.  The  vertical 
is  sometimes  chosen  for  titles  and  headings,  and  the  inclined  for  notes  and  dimen- 
sions. A  uniform  style  for  all  lettering  is  more  common. 

The  capitals  may  be  used  with  or  without  lower-case  (small)  letters,  or  with  small 
capitals  in  place  of  the  latter.  Titles  and  headings  are  usually  more  satisfactory 
if  all  capitals  are  used,  while  notes  are  more  easily  read  if  composed  of  capitals 
and  lower-case  letters. 

(a)  PROPORTIONS.     The   character,    proportions,    and   relations   of  the   ele- 
ments composing  the  letters  and  numerals  should  be  carefully  noted  and  fixed 
in  mind.     Examination  of  the  vertical  Gothic  will  show  that  W  is  the  widest 
character,  M  next,  and  A,  V,  and  4  next;  that  I,  J,  and  1  are  narrowest;  and  that 
the  others  are  nearly  equal  in  width   to  letter  O.      For  general  purposes  this 
width  should  be  from  f  to  £  of  the  height.     These  variations  are  designed  to 
overcome  the  appearance  of  inequality  which  would  result  if  all  were  made  equal 
when  used  in  words.     For  like  reason  A  is  extended  slightly  above  the  other 
letters  and  V  below.     The  curves  of  C,  G,  J,  0,  Q,  S,  U,  2,  3,  5,  6,  8,  and  9  also 
extend  slightly  beyond.     To  avoid  the  effect  of  top-heaviness  the  upper  part  of 
B,  C,  E,  G,  K,  R,  S,  X,  Z,  2,  3,  5,  6,  and  8  is  slightly  narrower  than  the  lower, 
and  certain  parts  in  B,  E,  F,  H,  R,  S,  X,  3,  5,  6,  and  8  come  slightly  above  the 
middle,  while  in  A,  G,  K,  P,  Y,  4,  and  9,  they  come  below.     All  curves  are  elliptic. 

In  practice  the  variations  in  height  and  wddth  noted  above  should  be  estimated 
by  eye. 

The  body  of  the  lower-case  letters  should  be  f  or  f  of  the  initial  capitals. 
The  sizes  of  letters  and  numerals  generally  suitable  are  indicated  in  Fig.  74. 

(b)  SPACING.     The  spaces  between  letters  vary  in  shape  with  each  different 
combination.     In  order  to  make  these  spaces  appear  equal  in  size  and  thus  avoid 
the  effect  of  crowding  or  isolating  letters,  it  is  necessary  to  increase  the  spaces 
in  certain  cases  and  decrease  them  in  others.     Thus,  considering  the  vertical 
capitals,  the  spaces  of  I  should  be  considerably  greater  than  of  other  letters, 
especially  when  parts  of  adjoining  letters  are  ||  to  it.      In  letters  whose  sides  are 
curved  as  in  O,  C,  D,  etc.,  it  is  generally  necessary  to  decrease  the  space.     This  is 
also  true  of  letters  having  oblique  sides  as  in  A,  V,  etc.,  and  letters  having  a 
greater  space  at  one  or  both  sides  as  in  L,  J,  P,  T,  etc. 

The  simplest  method  of  obtaining  good  spacing  is  to  sketch  the  words  lightly 
and  study  the  effect. 

The  space  between  words  should  be  about  1£  times  the  width  of  letter  0; 
that  between  sentences  in  line  about  three  O's;  and  that  between  ||  lines  of  letter- 
ing about  equal  to  the  height  of  the  shortest  letters  in  either. 


30  KSSKNTIAI.S  OF   MKCIIAMCAI.   DHAITINC 

(c)  PENCILING.     In  penciling  titles,  etc.,  and  notes,  rule  only  the  hor.  guide 
lines,  and  a  few  vert,  or  slant  lines  to  preserve  the  correct  positions.     For  method 
of  planning  a  title  see  Art.    102(c).      In    dimensioning  drawings   estimate   the 
heights  of  the  numerals  by  eye. 

Draw  all  characters  freehand,  using  a  fairly  sharp  pencil  point.     Art.  9. 

Before  lettering  a  drawing,  the  style  to  be  used  should  be  practiced  until 
it  can  be  done  reasonably  well.  Unless  otherwise  directed,  begin  with  the  vertical 
capitals,  observing  the  order  given  by  the  numbers  below  them-;  then  practice 
word-combinations;  then  the  figures. 

Place  Fig.  73  close  to  the  work  and  analyze  each  character.  A  convenient 
order  for  drawing  the  strokes  is  indicated  by  the  arrows.  To  determine  the  proper 
proportions  and  spacing,  first  point  the  forms  and  sketch  lightly  the  main  lines. 

Bear  in  mind  that  the  value  of  the  practice  is  in  the  carefulness  and  not  in  the 
amount.  Do  not  mix  the  styles. 

(d)  INKING.     Use   the    medium-point   writing    pen    for    small    letters   and 
figures,  and  the  coarse  pen  for  large  letters,  etc.     The  width  of  the  lines  should 
be  uniform  and  obtained  at  one  stroke. 

Observe  number,  order,  and  direction  of  strokes  as  in  penciling. 


I   !    I 


FIG. 73 


TITLE  AND    FILING    INDEX.        SPACE  ABOUT 


ICT   SPEED  LATHE 
HEAD   STOCK  DETAILS 

SMITH    M'FG   CO.     NEW  YORK 
Scale:    Full    Size  Date:   4-7-18 

Dr.  A.PS.     Tr.  F  L.       Ch.  H.M.      App. 


S.LrlO-5 


BILL  OF  MATERIAL. 

f* 

:NO. 

NAME 

REQ'D 

MAT. 

REMARKS 

7^- 

i  1 

Plain   Bearing 

1 

C.I. 

Pait.#B-5 

1 

L2 

St'd  Hex.  Bolts  i"x2" 

2 

W.I. 

C.H.H'ds  8cNuts 

3T 

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1                             7                                             y 

SUB  TITLES   AND    IDENTIFYING   MARK. 

FACE  PLATE  S.Lr8 

Scale:    Half  Size 
Make    2    C.I.    Patt.  *F-4 
FTA.O. 

NAMES   OF  VIEWS.  SECTION    PLANES.  ETC. 

"SECTION     PLAN     DIAGRAM     A— 


B 


NOTES,  FINISH    MARKS   AND    DIMENSIONS. 

!3Th.  4-  Holes  for  M-6.   |x|'Key 


'I1 


i -Bore 


FIG.  74 


CHAPTER  IV 
GEOMETRIC    CONSTRUCTION 


26.  Geometric    and    Practical    Methods.     The    accurate    execution    of    the 
views  and  diagrams  by  which  the  lines  and  surfaces  of  an  object  are  represented 
involves  the  graphic  solution  of  various  plane  problems  such  as  the  division  of 
lines  and  the  construction  of  perpendiculars,  parallels,  polygons,  etc. 

In  geometry  (Art.  2)  these  solutions  are  obtained  by  the  orderly  application 
of  geometric  principles,  which  require  the  use  of  the  compasses  and  a  straightedge 
only  in  drawing  the  necessary  working  or  construction  lines. 

While  it  is  frequently  necessary  in  practical  drafting  to  obtain  the  solution 
by  a  geometric  method,  the  draftsman  is  generally  enabled  to  shorten  the  con- 
structive process  or  to  obtain  a  direct  solution  by  means  of  the  T-square,  triangles, 
dividers,  etc. 

The  constructions  explained  in  this  Chap,  are  those  most  commonly  applied 
not  only  in  drafting,  but  in  the  laying  out  of  the  actual  lines  and  surfaces  of 
objects  by  the  workman. 

The  geometric  method  is  that  first  given.  Where  a  practical  method  (P.  M.) 
is  not  also  given,  the  geometric  method  would  ordinarily  be  used. 

27.  To  bisect  a  straight  line,  A-B,  or  a  circular 
arc,  A4B.    Fig.  75.     With  A  and  B  as  centers  and 
any  rad.  describe  arcs  intersecting  in  pts.  1  and  2. 
Through  1   and  2  draw  a  st.  line,  which  will  be  J_ 
to  A-B  and  bisect  it  at  3,  and  the  arc  A4B  at  4. 

Note  1. — Every  pt.  in  the  _L  bisector  1-2  is  equidistant  from 
A  and  B;  hence  a  different  rad.  may  be  taken  for  each  of  its 
determining  pts.  1  and  2,  or  both  located  upon  the  same  side  of 
A-B. 

Note  2. — The  J_  bisector  of  a  chord,  if  extended,  will  pass 
through  the  center,  5,  of  the  O  of  which  the  arc  A4B  is  a  part, 
and  will  bisect  the  O,  also  the  Z  A5B  of  which  the  arc  A-B  is 
the  measure. 

Note  3. — The  method  of  drawing  a  O  or  an  arc  of  given  rad. 
through  two  given  pts.,  or  upon  a  given  chord,  is  evident. 

P.  M.     Obtain  division  with  the  dividers,  Art.  15  (a) . 

28.  To  bisect  an  angle,  CAB.      Fig.  76.     With 
any  rad.  locate  pts.    1    and  2  upon  the  sides,  equi- 
distant from  vertex  A.     With  1  and  2  as  centers, 
any  rad.,  locate  an  equidistant  pt.  3.     Draw  3-A,  the 
bisector  of  the    Z . 

Note. — When  the  vertex,  A,  is  not  usable,  draw  ||s  to  the 
given  sides  (see  Art.  36),  obtaining  A',  and  bisect  Z  C'A'B'. 

33 


FIG  76 


34 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


Fife.  77 


29.  To  construct  an  angle  equal  to  a  given  angle,  CAB.     Fig.  77.     Draw  -an 
indefinite  line  A'-B'.     With  A  as  center  and  any  rad.,  draw  arc  C-B  to  cut  the 
sides  of  the  given  Z .     With  A'  as  center,  same  rad.,  draw  an  indefinite  arc  C'-B'. 
With  the  chord  of  C-B  as  rad.  and  center  B',  cut  C'-B',  at  C7.     Draw  A'-C'  com- 
pleting the  required  Z . 

30.  To  divide  a  straight  line,  A-B,  into  any  number  of  equal  parts  (say  5). 
Fig.  78.     Draw  A-4  at  any   Z  with  A-B.     Draw  B-l'  at  the  same    Z,  Art.  29. 
From  A  and  B  set  off  any  distance  upon  A-4  and  B-l',  as  many  times  as  the 
required  number  of  divisions  on  A-B,  less  one.     Draw  4-4',  3-3',  etc.,  which  will 
divide  A-B  as  required. 

SECOND  METHOD.  Set  off  five  equal  distances 
upon  A-5  from  A,  and  draw  ||s  to  5-B  by  making  Zs 
at  4,  3,  2,  and  1  equal  to  Z  A5B. 

Note. — Any  line,  as  1-1",  ||  to  one  side,  5-B,  of  a  A,  divides 
the  other  two  sides,  A-B  and  A-5,  proportionally.  Thus  the 
ratio  of  A-l"  to  1"-B  equals  the  ratio  of  A-l  to  1-5;  also  A-l" 
is  to  A-l  as  A-B  is  to  A-5. 

P.  M.  Draw  A-5  at  any  Z  with  A-B  and  from 
A  set  off  five  equal  distances  upon  it.  Draw  ||s  to 
5-B  through  4,  3,  2  and  1  by  Art.  8(c),  dividing 
A-B  as  required;  or  obtain  divisions  by  Art.  15(a). 

31.  To  divide  a  straight  line,  A-E,  into  parts 
proportional  to  those  of  a  given  divided  line,  A-D. 
Fig.  79.      The  method  is  evident  from  Art.  30. 

32.  To  lay  off  the  length  of  a  given  circular  curve 
upon  a  straight  line.      There  is  no  exact  method. 
Divide  the  given  curve  into  short  arcs,  not  neces- 
sarily equal,  whose  chords  will  closely  approximate 
the  lengths  of  the  subtended  arcs.     Lay  off  these 
chords  successively  upon  the  st.  line. 

Note. — The  length  of  the  circumference  of  a  O  is  3.1416 
times  the  diam.;  hence  it  may  be  computed  and  the  nearest 
fraction  taken  from  a  table  of  decimal  equivalents. 

33.  To  lay  off  the  length  of  a  given  straight  line 
upon  an  arc.     The  method  is  evident  from  Art.  32. 

34.  To  draw  a  perpendicular  to  a  line,   A-B, 
from  or  through  a  given  point  C. 

(a)  WHEN  C  is  UPON  A-B,  AT  OR  NEAR  THE  MIDDLE  OF  THE  LINE.     Fig.  80. 
Regard  A-B  as  a  st.   Z  with  vertex  at  C  and  proceed  as  in  Art.  28.     C-3  will  be 
the  required  _L . 

(b)  WHEN  C  is  OPPOSITE  OR  NEARLY  OPPOSITE  THE  MIDDLE  OF  A-B.     Fig.  81. 
Locate  pts.  1  and  2  equidistant  from  C,  and  pt.  3  equidistant  from   1   and  2, 
as  in  (a).     Draw  C-3,  the  required  J_. 

(c)  WHEN  C  is  UPON  A-B  AND  AT  OR  NEAR  THE  END  OF  THE  LINE.     Fig.  82. 
With  C  as  center,  any  rad.,  draw  an  arc  cutting  A-B  at  1.     With  1   as  center, 


GEOMETRIC  CONSTRUCTION 


same  rad.,  cut  this  arc  at  2.  With  2  as  center,  same 
rad.,  draw  an  arc  above  C.  Through  1  and  2  draw 
a  st.  line  to  cut  the  last  arc  at  3.  Draw  C-3,  the 
required  _]_ . 

Note  1. — The  dotted  arcs  suggest  another  method  of  locat- 
ing pt.  3. 

Note  2. — Any  Z,as  1C3,  inscribed  in  a  semicircle,  is  a  right  Z. 

SECOND  METHOD.  Fig.  82.  Assume  any  pt.  2, 
not  upon  A-B,  as  center,  and  with  rad.  2-C,  draw  an 
arc  cutting  A-B  at  1.  Through  1  and  2  draw  a  st. 
line  to  cut  this  arc  at  3.  Draw  C-3. 

(d)  WHEN  C  is  OPPOSITE  OR  NEARLY  OPPOSITE 
THE  END  OF  A-B.  Fig.  83.  From  C  draw  any  line 
C-l,  intersecting  A-B  obliquely.  Bisect  C-l  at  2. 
With  2  as  center,  rad.  2-C,  cut  A-B  at  3.  Draw  3-C, 
the  required  _1_. 

Note. — Compare  with  second  method  of  preceding. 

SECOND  METHOD.  Fig.  83.  From  any  two  pts. 
on  A-B,  as  1  and  A,  and  radii  1-C  and  A-C,  describe 
arcs  to  intersect  at  4.  Draw  C-4 . 

P.  M.     For  all  cases:  Obtain  the  _|_  by  Art.  8(d). 

35.  To  draw  a  line  at  an  angle  of  any  given 
magnitude  in  a  quadrant,  with  a  given  line,  A-B, 
at  A.  Fig.  84. 

(a)  45°.      Draw    A-l    at    90°     with     A-B     by 
Art.  34 (c).     Bisect  Z  1AB  by  Art.  28;  then  A-3  will 
make  an  Z  of  45°  with  A-B.     See  also  Note  1. 

(b)  60°.      With  A  as  center,  any  rad.,  draw  an 
arc  cutting  A-B  at  2.     With  2  as  center,  same  rad., 
cut  this  arc  at  4.     Draw  A-4,  which  will  make  an  Z 
of  60°  with  A-B. 

(c)  30°.      Determine    a  60°  Z    4AB    as  in    (b) 
and  bisect  it;  then  line  A-5  will  be  at  30°  with  A-B. 
See  also  Note  1. 

(d)  15°.     Determine  a  30°  Z    5AB,    as    in    (c) 
and  bisect  it;  then  A-6  will  be  at  15°  with  A-B. 

(e)  75°.      Determine  a  90°  Z    1AB    as   in     (a) 
and    a    60°  Z    4AB    as    in    (b).      Bisect  the   30°  Z 
1A4  and  A-7  will  be  at  75°  with  A  B. 

(f)  By  trisecting  the  15°  arcs  of  quadrant  1-2 
and  dividing  the  5°  arcs  thus  obtained  into  degrees, 
by   trial    (Art.   15(a)),    a   line   at  any   intermediate 
degree  may  be  determined.     Lines  at   Zs  involving 
fractions  of  degrees  may  be  determined  by  the  same 
general  methods. 


FIG  82 


FIG.  83 


FIG.  64 


Note  1. — The  dotted  lines  suggest  another  method  of  locating  pts.  3  and  5. 

Note  2. — Observe  that  the  rad.  of  a  O  is  equal  to  a  chord  of  £  of  its  circumference 


M 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


FIG.  89 


P.  M.  Lines  at  90°,  45°,  60°,  30°,  15°,  and  75* 
may  be  obtained  by  Art.  8(d).  For  lines  at  inter- 
mediate degrees  see  (f).  The  protractor  (Art.  12) 
may  be  used  for  all  cases  not  involving  fractions 
other  than  \°. 

36.  To  draw  a  parallel  to  a  given  straight  line, 
A-B. 

(a)  AT    A     GIVEN    DISTANCE,'    C-D,     FROM     A-B. 

Fig.  85.  With  any  two  pts.,  1  and  2,  on  A-B  as 
centers  and  C-D  as  rad.,  describe  arcs  on  the  same 
side  of  A-B.  At  1  draw  a  _L  to  A-B,  cutting  the  arc 
from  1  at  3.  With  3  as  center  and  rad.  1-2,  cut  the 
arc  from  2  at  4.  Draw  3-4,  the  required  ||. 

P.  M.  At  any  pt.,  1,  on  A-B  draw  a  _L  1-3  (Art. 
8(d)),  and  make  1-3  equal  to  C-D.  Through  3  draw 
3-4  ||  to  A-B.  Art.  8(c). 

(b)  THROUGH  A  GIVEN  POINT  3.     Fig.  85.     Lo- 
cate any  two  pts.,  1  and  2,  on  A-B.     With  1-3  as 
rad.  and  pt.  2  as  center  describe  an  arc  on  the  same 
side  of  A-B  as  pt.  3.     With  3  as  center  and  rad.  1-2 
cut  the  last  arc  at  4.     Draw  the  required  parallel  3-4. 

P.  M.     Draw  3-4  ||  to  A-B  by  Art.  8(c). 

37.  To  construct  a  triangle. 

(a)  WHEN   THE   SIDES  ARE  GIVEN.     Fig.   86. 
Make  A-B  equal  to  one  of  the  given  sides.     With  A 
and  B  as  centers  and  radii  equal  to-  the  second  and 
third  sides  draw  arcs  intersecting  at  C.     Draw  A-C 
and  B-C  to  complete  the  A- 

P.  M.  When  the  given  sides  are  equal  draw  the 
second  and  third  at  60°  with  the  first  by  Art.  8(d). 

(b)  WHEN  A  SIDE  A-B,  AND  THE  ANGLES  AT  A 
AND  B,  ARE  GIVEN.     Fig.  86.      Make  the  side  A-B 
and  Zs  CAB  and  ABC  equal  to  the  given  side  and 
Zs,  and  extend  A-C  and  B-C  to  meet  at  C. 

(c)  WHEN  A  SIDE  A-B,  THE  ANGLE  AT  B,  AND 

THE    ANGLE    OPPOSITE    A-B,     ARE    GIVEN.        Fig.     86. 

Find  the  angular  magnitude  at  A  by  subtracting 
the  sum  of  the  known  Zs  at  B  and  C  from  180°  and 
proceed  as  in  (b).  See  Fig.  20. 

38.  To  construct  an  isosceles  triangle  when  the 
base,  A-B,  and  vertex  angle,  ACB,  are  given.    Fig.  86. 
Find  the  angular  magnitudes  at  A  and  B  by  subtract- 
ing the  known  Z  from  180°  and  bisecting  the  remain- 
der;   then  proceed  as  in  Art.  37(b). 


GEOMETRIC  CONSTRUCTION 


37 


39.  To  construct  an  equilateral   triangle  when 
the  altitude,  A-B,  is  given.     Fig.  87.     Draw  B-l  and 
B-2  at  30°  with  A-B,  Art.  35(c).     Draw  1-2  _|_  to 
A-B,  Art.  34(c). 

P.  M.      Draw  B-l   and   B-2    at  30°   with   A-B, 
Art.  8(d).     By  same  Art.  draw  1-2  _|_  to  A-B. 

40.  To  circumscribe  a    circle    about    a    given 
triangle,  ABC.      Fig.  88(a).    Draw  the  _L  bisectors 
of   either  two  sides,   as   A-B   and  B-C  (Art.  27)  to 
intersect  at  1;  pt.  1,  being  equidistant  from  A,  B,  and 
C,  will  be  the  center  of  the  required  O . 

Note  1. — The  center  of  any  O  is  at  the  intersection  of  the 
_L  bisectors  of  any  two  of  its  non-parallel  chords. 

Note  2. — The  method  of  drawing  a  O  through  any  three  pts., 
not  in  the  same  st.  line,  is  evident. 

41.  To  inscribe  a  circle  within  a  given  triangle, 
ABC.     Fig.  89.     Draw  the  bisectors  of  either  two  of 
the  Zs,  as  CAB  and  ABC  (Art.  28)  to  intersect  at  1, 
the  center  of  the  required  O  •     The  _L  distance  from 
the  center  to  any  side  is  its  rad. 

42.  To  inscribe  an  equilateral   triangle    within 
a  circle.     Fig.  90.     Draw  a  diam.  1-2.     From  either 
end,  2,  draw  chords  at  30°  with  1-2  by  Art.  35(c). 
Draw  3-4  to  complete  the  A. 

P.  M.  Draw  diam.  1-2.  Draw  sides  2-3  and  2-4 
at  30°  with  1-2  by  Art.  8(d)  and  join  3  and  4. 

43.  To  circumscribe  an  equilateral  triangle  about  a  circle.      Fig.  91.     Draw 
a  diam.  1-2.     From  2,  with  a  rad.  equal  to  that  of  the  given  O,  cut  the  diam. 
extended  at  3,  and  the  given  O   at  4  and  5.      Draw  3-4  and  3-5.      With  C  as 
center,  rad.  C-3,  cut  3-4  and  3-5  extended  at  6  and  7.     Draw  7-6  to  complete 
the  A. 

P.  M.     Draw    7-6   tangent   to   the    O   by   eye;  through  center  C  draw  7-4 
and  5-6  at  30°  with  7-6,  Art.  8(d) .     By  same  Art.  draw  7-3  and  3-6  at  60°  with  7-6. 

44.  To  construct  a  parallelogram  when  two  sides,  A-B   and   A-C,  and  the 
included  angle,  CAB,  are  given.     Fig.  92.     Make  the  sides  A-B  and  A-C  and 
the  included  Z  CAB  equal  to  the  given  sides  and  Z .      Draw  C-l  |  [  and  equal  to 
A-B  by  Art.  36 (b)  and  draw  B-l  to  complete  the  parallelogram. 

P.  M.      Make  Z    CAB  equal  to  the  given  Z    by  Art.  29.      Make  A-B  and 
A-C    equal  to  the  given  sides.     Draw  the  ||  sides  by  Art.  8(c). 

45.  To  construct  a  square  on  a  given  side,  A-B.     No  figure.     Draw  a  second 
side  A-C  _1_  to  A-B,  Art.  34(c).     Draw  C-l  ||  to  A-B,  and  B-l  ||  to  A-C,  by   Art. 
36(b),  to  complete  the  square. 

P.  M.     Draw  A-C  and  B-l  at  90°  with  A-B,  and  A-l  at  45°,  by  Art.  8(d). 
Draw  C-l  ||  to  A-B  by  Art.  8(c). 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


46.  To  inscribe  a  square  within  a  circle.  Fig.  93.  Draw  a  diam.  1-2. 
From  1  and  2  draw  chords  at  45°  with  1-2  (Art.  35(a))  to  form  the  square. 

P.  M.  Draw  _L  diams.  1-2  and  3-4.  At  45°  with  these,  draw  sides  1-4,  3-2, 
etc.,  Art.  8(d). 


47.     To  construct  a  square  on  a  given  diagonal,  1-2. 
is  evident  from  Art.  46. 


Fig.  93.     The  method 


FJG.  93 


FIG.  94 


FIG. 96 


48.  To    circumscribe    a    square    about    a    circle. 
Fig.  94.     Draw  _L  diams.  1-2  and  3-4.      Through  1,  2, 
3,  and  4  draw  ||s  to  these  diams.  (see  Art.  36(a)),  form- 
ing the  required  square. 

P.  M.  Draw  ||  tangents  8-7  and  5-6,  Art.  8(c). 
Through  the  center  and  at  45°  with  these  draw  5-7, 
Art.  8(d).  By  same  Art.  draw  tangents  J_  to  the  first 
two. 

49.  To  inscribe  a  regular  pentagon  within  a  circle. 
Fig.  95.      Draw  a  diam.  1-2  and  a  _L  rad.  3-4.      Bisect 
3-2  at  5.     From  5,  rad.  5-4,  cut  the  diam.  1-2  at  6. 
The  chord  of  4-6  is  equal  to  one  side  of  the  required 
pentagon.      Hence,  with  4  as  center,  rad.  4-6,  cut  the 
circumference  at  7  and  8.     With  7  and  8  as  centers, 
cut  it  again  at  9  and  10.     Draw  chords  7-4,  4-8,  etc., 
to  form  the  pentagon. 

P.  M.  Divide  the  circumference  into  five  equal 
parts  by  trial  (Art.  15  (a)),  and  connect  the  pts. 

50.  To  construct  a  regular  hexagon  on   a   given 
side,  A-B.     Fig.  96.      With  A  and  B  as  centers,  rad. 
A-B,  draw  arcs  intersecting  at  1.     With  1  as  center, 
same  rad.,   describe  a  O  cutting  the  first  arcs  at  2 
and  3.     With  2  and  3  as  centers,   same  rad.,  cut  the 
circumference  at  4  and  5.     Draw  chords  2-4,  B-3,  etc., 
to  complete  the  hexagon.     See  Art.  35,  Note  2. 

P.  M.  Draw  A-5  and  4-B  at  60°  with  A-B,  Art.  8(d). 
Through  their  intersection,  l,draw  2-3  parallel  to  A-B, 
Art.  8(c).  Draw  B-3,  2-A,  2-4,  and  5-3  at  60°  with 
2-3,  and  4-5  parallel  to  2-3. 

51.  To  inscribe  a  regular  hexagon  within  a  circle. 
Fig.  97.     Draw  a  diam.  1-2.     Draw  chords  1-3,  5-2, 
6-2,  and  1-4  at  60°  with   1-2,  Art.  35(b).     Draw  3-6 
and  4-5  to  complete  the  hexagon. 

P.  M.  Draw  diam.  1-2.  Draw  diam.  4-6  and  sides 
1-3,  5-2,  6-2,  and  1-4  at  60°  with  1-2,  Art.  8(d).  Draw 
sides  3-6  and  4-5  ||  to  1-2,  Art.  8(c). 

52.  To  construct  a  regular  hexagon  on  a    given 
long  diagonal,  1-2.     The  method  is  evident  from  Art.  51. 


GEOMETRIC  CONSTRUCTION 


53.  To  inscribe  a  regular  octagon  within  a  circle. 
Fig.  98.     Draw  J_  diams.  1-2  and  3-4.     Bisect  the  90° 
Zs  by  Art.  28.     Draw  chords   1-5,   5-4,  etc.,  to  form 
the  octagon. 

P.  M.  Draw  _L  diams.  1-2  and  3-4,  and  diams.  5-6 
and  7-8  at  45°  with  these,  Art.  8(d).  Draw  sides  1-5, 
5-4,  etc. 

54.  To  circumscribe  a  regular  octagon    about    a 
circle.      Fig.  99.      Circumscribe  a  square  about  the  O , 
Art.  48.     With  pts.  1,  2,  3,  and  4  as  centers,  rad.  1-C,  cut 
the  sides  at  S,  T,  U,  V,  etc.     Join  S,  X,  T,  W,  etc.,  to 
complete  the  octagon. 

P.  M.  Draw  ||  tangents  4-3  and  1-2.  _|_  to  these 
draw  1-2  and  2-3.  At  45°  with  these  draw  S-X,  T-W, 
Y-V,  and  Z-U. 


55.     To  construct  any  regular  polygon. 

METHODS. 


GENERAL 


FIG.  99 


FIG.  100 


FIG.  101 


(a)  WHEN  A  SIDE,  A-B,  is  GIVEN.     Fig.  100.      With  A  or  B  as  center,  rad. 
A-B,  describe  a  semicircle  upon  A-B  extended.      Divide  this  arc  by  trial  (Art. 
15(a))   into  as   many  equal  parts  as  the  required  polygon  has  sides   (say  7). 
Through  the  center,  B,    and  the  second    pt.    of   division,  2,    draw  B-2,  which 
will  be  a  second  side  of  the  polygon.     Describe  a  O  through  A,  B,  and  2.     (See 
Art.  40,  Notes  1  and  2.)     Apply  A-B  as  a  chord  from  2,  U,  V,  etc.,  and  draw 
2-U,  U-V,  etc.,  to  complete  the  polygon. 

Note  1.— Vertices  U,  V,  W,  and  X  may  also  be  determined  by  radials  from  B,  through  3,  4,  5, 
and  6,  as  shown. 

Note  2. — Line  B-l  would  be  the  second  side  of  a  polygon  of  twice  as  many  sides. 

(b)  WHEN  THE  CIRCUMSCRIBING  CIRCLE  is  GIVEN.     Fig.  101.      Draw  a  diam. 
1-2,  and  a  tangent  3-4  _|_  to  it.     With  1  as  center  and  any  rad.,  preferably  1-C, 
describe  a  semicircle   upon   3-4.     Divide  this  arc  and  draw  radials  as  in  (a). 
Join  pts.  U,  V,  W,  etc.,  thus  found,  to  complete  the  polygon. 

P.  M.     Divide  the  circumference  by  trial  (Art.  15  (a)),  and  join  the  pts.  I,  U, 
V,  W,  etc. 


•10 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


FIG    101 


(c)  WHEN  THE  INSCRIBED  CIRCLE  is  GIVEN.  Fig. 
101.  Divide  the  circumference  as  in  (b)u  Through 
these  pts.  of  division  draw  radials  C-W,  C-X,  etc.,  and 
obtain  one  side  of  the  inscribed  polygon,  as  X-W. 
Draw  a  tangent  S-T  ||  to  X-W,  which  will  be  one  side 
of  the  required  polygon.  The  method  of  obtaining 
the  others  is  evident. 

P.  M.  Divide  the  circumference,  as  in  P.  M.  of  (b); 
then  proceed  as  above. 

56.  To   construct   a  polygon   similar   to   a   given 
polygon,  ABCDE,  upon  a  given  side,  A'-B'.     Fig.  102. 

(a)  WHEN  A'-B'  is  EQUAL  TO  THE  CORRESPONDING 
SIDE,  A-B.     Draw  A'-B' equal  to  A-B.     Locate  vertices 
C',  D',  and  E'  by  drawing  intersecting  arcs  with  cen- 
ters A'  and  B'  and  radii  equal  to  the  distances  of  C,  D, 
and  E  from  A  and  from  B  respectively. 

SECOND  METHOD.  Divide  the  given  polygon  into 
As  by  diagonals  A-C  and  A-D.  Construct  Zs  at  A' 
and  B'  equal  to  those  at  A  and  B,  and  at  C'  and  D', 
equal  to  those  at  C  and  D  (Art.  29);  then  the  cor- 
responding As  of  the  given  and  required  figures  will  be 
similar  and  in  this  case  equal. 

P.  M.  Determine  the  vertices  by  Art.  57  and  join 
them. 

(b)  WHEN  A'-B'  is  GREATER  OR  LESS  THAN  A-B. 
Draw  A'-B'  equal  to  the  given  length  and  proceed  as 
in  (a),  second  method.     Then  the  corresponding  sides 
of  the  given  and  required  figures  will  be  proportional. 

Note  1. — It  is  sometimes  convenient  to  lay  off  A'-B'  upon  A-B 
or  A-B  extended,  and  then  draw  |  |s  to  the  other  sides,  as  indicated 
by  the  dotted  lines. 

Note  2. — When  the  ratio  of  A'-B'  to  A-B  is  given  (as  say  3  to  4) 
the  length  of  A'-B'  may  be  determined  by  a  scale  of  proportional 
lengths,  Fig.  (a).  Draw  indefinite  lines  A-X  and  A-Y  at  any  Z 
with  each  other.  On  one,  as  A-Y,  set  off  any  length,  A-l,  and  on 
the  other  A-2,  in  this  case  equal  to  J  of  A-l.  On  A-Y  set  off  the 
length  of  A-B.  Draw  2-1,  and  B'-B  ||  to  it;  then  A-B'  will  be 
equal  to  f  of  A-B  and  thus  be  the  required  length  of  A'-B'.  See 
Art.  30,  Note. 

57.  To  plot  a  figure  similar  to  a  given  figure  by 
means  of  a  base  line  or  center  line,  and  offsets  (in 
this  case    an    irregular    curve   A-F).     Fig.     103.      In 
any   convenient   position   with   respect   to    the   given 
figure  draw  base  line  X-Y.     From  A,  F,  and  any  num- 
ber  of   intermediate   pts.,   draw   offsets    _L    to    X-Y. 
Having  thus  referred   a  sufficient  number  of  limiting 


GEOMETRIC  CONSTRUCTION 


41 


pts.,  draw  X'-Y'  as  the  line  of  reference  for  the  re- 
quired figure.  Upon  X'-Y'  set  off  l'-2',  l'-3',  etc., 
equal  to  1-2,  1-3,  etc.,  and  draw  offsets  l'-A',  2'-B', 
3'-C',  etc.,  equal  to  1-A,  2-B,  3-C,  etc.  Through  pts. 
A',  B',  C',  etc.,  thus  determined,  draw  the  required 
curve.  (See  Art.  18(a).)  Note  application  of  princi- 
ple in  Fig.  (a),  also  in  Fig.  132 (a),  (b). 

(a)       TO    DRAW    THE    FIGURE     TO    AN    ENLARGED     OR 

REDUCED  SCALE  (say  enlarged  in  ratio  3  to  2).  The 
co-ordinate  distances  would  be  f  =  1^  times  those  of  the 
corresponding  pts.  of  the  given  figure,  and  may  be 
obtained  by  Art.  56(b),  Note  2;  or  proportional  dividers 
may  be  used. 

58.  To  draw  a  tangent,  A-B,  to  a  circle,  through  a 
given  point  A. 

(a)  WHEN  A  is  ON   THE    CIRCUMFERENCE.      Fig. 
104(a).      Draw  rad.   C-A,  and   tangent  A-B  _L   to  it, 
Art.  34(c). 

P.  M.     Draw  A-B  JL  to  rad.  C-A  by  Art.  8(d). 

(b)  WHEN  A  is  ON  THE  CIRCUMFERENCE  AND  THE 
CENTER  INACCESSIBLE.   Fig.  104(b).     With  A  as  center, 
any  rad.,  cut  the  curve  at  1  and  2.     Draw  chord  1-2. 
Draw  A-B  ||  to  1-2  by  Art.  36(b). 

P.  M.      Obtain  chord  1-2  as  above.      Draw  A-B  || 
to  it  by  Art.  8(c). 

(c)  WHEN  A  is  OUTSIDE  OF  THE  CIRCUMFERENCE. 
Fig.    105.      Join  A  to  center  C.     Upon  A-C  draw  a 
semicircle  to  determine  pt.  of  tangency  B.     Draw  tan- 
gent A-B.     See  Art.  34 (c),  Note  2. 

Note. — A  second  tangent,  A-B',  may  be  drawn,  as  indicated. 

P.  M.     Draw  A-B  tangent  to  the  O  by  eye.   Locate 
pt.  of  tangency  by  a  rad.  C-B  JL  to  A-B,  Art.  8(d). 

59.  To  draw  a  tangent,  A-B,  to  two  given  circles. 
Fig.  106.     Draw  line  of  centers  1-2.     Subtract  length  of 
rad.   of  smaller  O  from  that   of  the  larger  and  draw 
concentric  arc  3-4.     Obtain  tangent  2-3  by  Art.  58(c). 
Draw  rad.   1-A  through  3,   and  make  Z  B  2  5  equal 
to  Z  A  1  4,  Art.  29.  Draw  A-B,  the  required  tangent. 

Note. — To  draw  a  tangent,  C-D,  passing  between  the  centers — 
add  length  of  rad.  of  smaller  Q  to  that  of  the  larger,  and  draw  con- 
centric arc  6-7.  Draw  tangent  2-7,  and  rad.  1-7,  locating  pt.  C. 
Make  Z  D  2  1  equal  to  Z  2  1  7,  thus  locating  D.  Draw  C-D. 

P.  M.  Draw  tangents  A-B  and  C-D  by  eye.  Locate 
pts.  of  tangency  as  in  P.  M.  of  Art.  58(c). 


FIG.  104 
(a) 


A       (b) 


FIG.  105 


«a 


ESSENTIALS  OF  MECHANICAL  DKAlTINd 


60.  To  draw  a  circular  curve  of  given  radius,  D-E,  tangent  to  a  given  circular 
curve  and  to  a  given  straight  line,  A-B.     Fig.  107.     At  distance  D-E  from  A-B 
draw  a  ||  1-2.      Upon  any  rad.,  C-3  extended,  set  off  3-4  equal  to  D-E.      Con- 
centric with  given  curve,  and  with  rad.  C-4,  draw  an  arc  cutting  1-2  at  5,  the 
center  of  the  required  curve.     Line  C-5  and  a  _L  to  A-B  from  5  determine  pts.  of 
tangency  7  and  6.      See  application  in  Fig.  (b)  at  G. 

Note. — The  method  of  drawing  a  circular  curve  of  given  rad.  tangent  to  two  given  circular  curves 
is  evident  from  Fig.  (a). 

61.  To  draw  a  circular  curve  tangent  to  two  given  straight  lines,  A-B  and 
C-D.  Fig.  108. 

(a)  WHEN  A  POINT  OF  TANGENCY,  E,  ON  A-B  is  GIVEN.  Fig.  (a).  Extend 
A-B  and  C-D  to  intersect  at  1,  and  make  1-E'  equal  to  1-E.  At  E  and  E' 
draw  _Ls  to  intersect  at  3,  the  center  of  the  required  curve;  or  bisect  the  included 
Z  BID  and  draw  ±  at  E  to  cut  the  bisector  1-2  at  3. 

Note  1. — When  the  vertex,  1,  is  inaccessible,  the  bisector  may  be  obtained  by  Art.  28  Note. 

Note 2. — The  construction,  when  A-B  and  C-D  are  ||,  or  at  right  Zs,  is  evident  from  Figs,  (b) 
and  (c). 


FIG  109 


FIG.  107 


FIG    108 


(b)  WHEN  THE  RADIUS,  3-E,  is  GIVEN.  Fig.  (a). 
At  distance  3-E  draw  ||s  to  A-B  and  C-D,  intersecting 
at  center  3;  or  obtain  bisector  1-2  as  in  (a)  and  a  ||  to 
one  side  intersecting  1-2  at  3.  See  application  in  Fig. 
107(b)  at  F. 

Note. — When  A-B  and  C-D  are  at  right  Zs,  the  rad.  may  be  ap- 
plied as  in  Fig.  (c). 


FlG    110 


62.     To  draw  a  circular  curve  tangent  to  three  straight  lines,  A-B,  A-C, 
and  C-D.     Fig.  109.     The  construction  is  identical  with  that  of  Art.  42. 


GEOMETRIC  CONSTRUCTION 


43 


63.  To  draw  circular  curves  tangent  to  two  given  parallel  straight  lines, 
A-B  and  C-D,  at  B  and  C,  and  to  each  other  at  any  point  E,  in  line  B-C.     Fig.  110. 
Draw  _L  bisectors  of  B-E  and  E-C.     At  B  and  C  draw  J_s  to  A-B  and  C-D  to 
cut  the  bisectors  at  1  and  2,  the  centers  of  the  required  arcs. 

64.  To  draw  an  ellipse  when  the  axes,  A-B  and  C-D,  are  given. 

(a)  BY  FOCAL  RADII.     Fig.  111.     Draw  the  axes   J_  to  each  other  at  their 
middle  pts.      With  either  end,  C,  of  the  minor  axis  as  center  and  rad.  equal  to 
i  of  the  major  axis,  cut  the  latter  at  F  and  F',  the  foci  of  the  ellipse.     Between 
the  center,  E,  and  either  focus  place  any  pt.,  1.     With  the  foci  as  centers  and 
rad.  A-l,  draw  arcs  upon  opposite  sides  of  A-B.     With  same  centers  and  rad. 
B-l ,  cut  these  arcs  at  2, 3,  4,  and  5,  which  will  be  pts.  of  the  required  curve.    Assume 
a  sufficient  number  of  other  pts.  for  focal  radii 

on  A-B  and  proceed  in  like  manner.  Draw  the 
curve  through  the  pts.  thus  found,  freehand. 
See  Art.  18(a). 

Note. — The  method  of  obtaining  a  tangent,  T-T',  at  a 
given  pt.,  2,  in  the  curve  is  evident.     See  Art.  2(d). 

(b)  BY  TRAMMEL  METHOD.     Fig.  111.    On 
the  st.  edge  of  a  piece  of  paper  mark  off   G-I 
equal  to  |   of  the  major  axis,  and   I-H  equal 
to  \  of  the  minor.      Moving  this  trammel  so 
that  pt.  G  remains  on  the  minor  axis  and  H  on 
the  major,  set  off  a  sufficient  number  of  pts.  for 
the  successive  positions  of  I  to  determine  the 
required  curve. 

Note. — Having  located  pts.  for  J  of  the  curve,  corre- 
sponding pts.  could  be  determined  by  Art.  57. 

(c)  BY  REVOLUTION  OF  A  CIRCLE.    Fig.  112. 
Upon  the  axes  describe  O  s.      Divide  these  O  s 
into   the   same  proportional   parts   by    diams. 
If  the  large  O  be  imagined   to  revolve  about 
axis  A-B,  pts.  1,  2,  etc.,  will  appear  to  move_L 
to  A-B.     When  1  and  6  coincide  with  D  and 
C,  pts.  2,  3,  etc.,  will  have  moved  proportional 
distances  which  are  determined  by  ||s  to  A-B 
from  the  corresponding  pts.  2',  3',  etc.,  of  the 
smaller  O ,  at  2",  3",  etc.,  of  the  required  curve. 

Note. — The  figure  indicates  a  second  method  of  obtain- 
ing a  tangent  at  a  given  pt.  in  the  curve. 

(d)  BY  PARALLELOGRAM  METHOD.  Fig.  113. 
Draw  ||s  to  A-B  and  C-D   through  A,    B,  C, 
and  D,  forming  a  parallelogram.      Divide  A-F 
into  any  number  of  equal  parts  and  A-E  into 
the  same    number  of    equal  parts.      Through 

the  pts.  of  division  on  A-F,  draw  D-l,  D-2,  D-3.      Through  the  corresponding 
pts.  on   A-E  draw    C-l,   C-2,   C-3    intersecting  the  lines  from  D  at  1',  2',  3', 


44 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


which  will  be  pts.  in  the  required  ellipse.  In 
like  manner  find  pts.  for  remainder  of  curve. 

Note  1 . — The  same  construction  applies  when  any  two 
conjugate  diameters  are  given. 

Note  2. — The  method  of  inscribing  an  ellipse  in  any 
given  parallelogram  (not  square)  is  evident. 

(e)  BY  CIRCULAR  ARCS.  Fig.  114.  The 
following  is  one  of  several  methods  of  approx- 
imating an  ellipse:  Draw  D-B.  Make  1-2 
equal  to  1-D  and  D-3  equal  to  2-B.  Draw 
a  _l_  bisector  to  3-B,  cutting  C-D  extended  at 
4,  and  A-B  at  5.  Make  1-5'  equal  1-5  and  1-4'  equal  1-4,  and  draw  4-6',  4'-6", 
and  4-6'".  With  5  and  5'  as  centers,  rad.  5-B,  draw  arcs  6-6'"  and  6"-6';  then 
with  4  and  4'  as  centers,  rad.  4-D,  draw  arcs  6'-6  and  6"'-6"  to  complete  the 
curve. 


FIG    114- 


CHAPTER  V 


ORTHOGRAPHIC   PROJECTION 


65.  General  Principles.  It  has  been  noted  that  mechanical  drawings  are 
made  for  the  purpose  of  showing  the  exact  facts  of  form,  dimension,  and  arrange- 
ment of  parts  in  objects  of  a  structural  character.  (See  Art.  1.)  To  express  these 
facts  fully  and  clearly  it  is  necessary  to  represent  the  object  by  two  or  more 
related  drawings  each  of  which  gives  certain  information  that  the  others  lack. 

These  drawings,  though  made  upon  one  plane,  the  paper,  by  the  methods  of 
Chap.  IV,  are  regarded  as  projections  of  the  image  of  the  object  upon  different 
planes  ||  to  the  axes  or  principal  dimensions  of  the  object,  and  imagined  to  be 
obtained  by  means  of  _Ls  to  those  planes  from  all  pts.  of  the  object;  that  is,  in 
accordance  with  the  principles  of  Orthographic  (true  drawing)  Projection. 

(a)  In  Fig.  115(a),  ABCD  is  a  pictorial  drawing  of  a  hollow  rectangular 
block,  placed  squarely  in  front,  with  the  center  of  the  opening  on  the  level  of  the 
eye.     By  experiment  with  a  similar  object,  it  will  be  observed  that  the  front  sur- 
face, being  at  right  Z  s  to  the  direction  in  which  it  is  seen,  appears  in  its  exact  form. 

The  surfaces  of  the  opening,  although  at  right  Z  s  with  the  front,  appear  fore- 
shortened and  to  incline  towards  each  other;  the  farther  lines  appear  shorter  than 
the  nearer,  and  the  receding  lines  to  incline  and  converge  towards  a  pt.  at  the  center. 

The  apparent  decrease  in  size, 
foreshortening,  inclination,  and  con- 
vergence is  due  to  the  position  of  the 
lines  and  surfaces  of  the  object  with 
respect  to  the  eye  of  the  observer,  and, 
obviously,  if  the  position  of  the  eye  be 
changed,  the  appearance  will  also 
change.  Thus,  if  the  eye  be  moved 
a  certain  distance  upward,  the  object 
would  appear  as  shown  in  Fig.  (b) 
and  if  moved  upward  to  the  right,  as 
in  (c). 

It  is  evident  that  none  of  these 
pictorial  representations  shows  the 
exact  form,  size,  and  relation  of  all 
the  lines  and  surfaces. 

(b)  Let    LMNO    (Fig.     115(a)) 
represent    a  vert,   pane  of  glass  or 
wire  screen  placed    ||    to   the  front 
surface    of    the     object.       Consider 
this    glass    or   screen   to    be    merely 


(a) 


FIG.  115 


(c) 


45 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


a  plane,  and  the  picture  ABCD,  as  obtained  by  tracing  lines  upon  the 
plane  to  exactly  cover  the  outlines  of  the  object  seen  through  it.  Since  the  plane 
is  nearer  to  the  eye  than  the  object,  and  the  rays  of  light  converge  from  the 
object  to  the  eye,  it  follows  that  all  lines  of  the  tracing  must  be  shorter  than  those 
of  the  object;  also,  that  if  the  object  be  projected  forward  until  its  front  surface 
coincides  with  the  plane,  the  tracing  of  that  surface  would  be  identical  both  in 
form  and  dimensions  with  the  surface  itself. 

(c)  If  now,  instead  of  moving  the  object,  lines  be  imagined  to  extend  from 
all  its  pts.,  A,  B,  C,  D,  etc.,  _L  to  the  plane,  as  shown  picto.rially  in  Fig.  116,  these 
_Ls  would  intersect  the  plane  at  Av,  Bv,  Cv,  Dv,  etc.  These  pts.  are  called 
projections  of  the  pts.  of  the  object,  and  lines  AV-BV,  BV-CV,  etc.,  joining  them,  will 
be  projections  of  the  lines  and  surfaces  of  the  object. 

The  plane  is  a  plane  of  projection  and  the  _Ls  are  the  projectors  of  the  pts. 


u.           WIDTH          __j 
OR  BREADTH^] 

D* 

E» 

C» 

rv 

t 

o 

I 

1 

FIG. 116 


AVHV  B»GV 

Fio.117 


M 


(d)  Since  the  projectors  are  ||  to  each  other,  and  the  rear  pts.  of  the  object 
are  perpendicularly  back  of  the  front  pts.,  it  follows  that  the  projections  of  the 
front  and  rear  pts.  will  coincide,  as  indicated  in  the  figure.     The  projection,  when 
viewed  squarely,  as  in  Fig.  117,  thus  shows  what  the  eye  would  see  if  imagined  to 
be  directly  opposite  each  pt.  of  the  object  at  the  same  time;  namely,  the  exact 
form  and  dimensions  of  the  front  and  rear  surfaces,  and  the  dimensions  of  the 
object  from  left  to  right  and  bottom  to  top. 

(e)  Although  in  this  case  the  first  dimension  is  the  length  and  the  latter  the 
width,  it  is  convenient  in  speaking  of  the  dimensions  of  an  object  in  a  definite 
position  to  call  the  hor.  dimension  from  left  to  right  the  width,  or  breadth,  that 
from  front  to  back  the  depth,  and  the  vert,  dimension  the  height,  regardless  of 
their  extent. 

In  the  notation  of  this  proj.  and  others  which  will  be  explained,  the  first  letter 
in  each  instance  indicates  the  pt.  of  the  object  nearer  the  plane,  and  the  small 
letter  the  plane  upon  which  the  proj.  of  that  pt.  lies.  To  avoid  confusion,  the 
notation  of  the  opening  is  omitted. 

(f)  Since  the  three  dimensions  of  an  object  are  JL  to  each  other  and  a  plane 
has  but  two  dimensions,  it  follows  that  when  two  dimensions  of  the  object  are 


ORTHOGRAPHIC  PROJECTION 


47 


projected  in  their  exact  size  the  remaining  dimension  is  not  seen;  that  is,  no  more 
than  two  dimensions  can  be  shown  in  their  exact  size  and  relation  in  one  proj. 

Hence,  to  show  the  dimension  from  front  to  back  (depth),  another  proj.  upon 
a  plane  _l_  to  the  front  or  vertical  plane  must  in  like  manner  be  obtained. 

Thus  in  the  pictorial  illustration  Fig.  118,  DHCHFHEH,  etc.,  is  the  proj.  of 
the  object  upon  a  top  or  horizontal  plane  ONQP,  and  BPGPFPCP,  etc.,  its  proj. 
upon  a  side  vertical  or  profile  plane 
MRQN,  at  right  Zs  to  the  first  two. 
Note. — The  relation  of  the  planes  may  be 
illustrated  by  means  of  a  paper  box,  hinged 
panes  of  glass,  or  screens. 

(g)  To  show  these  projs.  as  they 
are  generally  arranged  in  a  mechan- 
ical drawing,  the  planes  of  two  of  the 
projs.  are  imagined  to  be  revolved 
about  their  lines  of  intersection  into 
the  plane  of  the  other  regarded  as  the 
plane  of  the  drawing,  as  in  Fig.  119. 

(h)  It  should  be  noted  that  the 
projs.  .upon  the  top  and  side  planes 
are  precisely  the  same  as  would  be 
obtained  upon  the  front  plane  if  the 


FiG.118 


A" 

CH 

B" 

T 


h 


G.L. 


WIDTH 


Mwiuin        ^ 
OR   BREADTHJ 

ill L 


V 


T 


HTorP I 


J-DEPTH*! 


43  ESSENTIALS  OF  MECHANICAL  DRAFTINC 

object  were  turned  from  the  position  given  so  that  its  dimensions  of  width  and 
depth,  and  then  its  height  and  depth,  are  ||  to  that  plane. 

The  opening  cannot  be  seen  when  the  object  is  looked  at  squarely,  either  from 
above  or  from  the  side,  but  as  it  is  necessary  to  show  its  projs.  upon  the  top  and 
side  planes, — the  lines  whose  projs.  do  not  coincide  with  those  of  visible  lines 
are  represented  by  dashed  lines,  to  indicate  that  they  are  invisible  or  hidden. 
Observe  that  the  outer  lines  of  all  projs.  represent  visible  lines  of  the  object;  those 
within  represent  hidden  lines  when  seen  in  the  other  projs.  to  lie  below  or  behind 
some  solid  portion. 

(i)  It  is  evident  that  the  three  dimensions  are  shown  by  any  two  of  the 
projs.;  that  two  _L  projs.  are,  therefore,  necessary  to  show  the  three  dimensions; 
and  that  the  three  projs.  together  determine  completely  and  clearly  the  exact 
form,  size,  and  relation  of  all  lines  and  surfaces  of  the  object. 

(j)  These  three  mutually  _L  planes  of  proj.  are  called  the  co-ordinate  planes, 
and  for  brevity  are  denoted  by  V,  H,  and  P  respectively. 

A  proj.  is  named  from  the  plane  upon  which  it  is  imagined  to  be  obtained,  not 
from  the  particular  part  seen  or  shown.  Thus  those  upon  V,  H,  and  P  are  respec- 
tively the  vertical,  horizontal,  and  profile  projection. 

In  practical  drafting,  the  first  is  called  a  front  elevation  or  front  view,  the  second 
a  plan  or  lop  view  and  the  third  a  side  elevation  or  side  view. 

The  terms  "plan"  and  "elevation"  are  used  chiefly  in  architectural  drafting; 
the  term  "view"  is  more  generally  used. 

(k)  The  line  of  intersection  of  V  and  H  is  called  the  ground  line  or  trace  of 
V  and  H,  and  is  denoted  by  G  L. 

When  looking  squarely  at  V,  the  G  L  represents  H;  and  when  looking  at  H,  it 
represents  V.  The  line  of  intersection  of  V  and  P  is  called  the  vertical  trace  of  P; 
that  of  H  and  P  is  the  horizontal  trace  of  P.  These  are  denoted  by  V  T  of  P  and 
H  T  of  P. 

When  looking  at  V  or  H,  these  traces  indicate  the  position  of  P.  When  looking 
at  P,  they  represent  V  and  H.  The  verts,  and  hors.  from  the  views  to  the  traces 
represent  the  projectors  to  the  planes. 

(1)  When  H  and  P  are  revolved,  the  front  and  top  views  of  any  pt.  are  seen 
to  lie  in  the  same  vert.;  the  front  and  side  views  in  the  same  hor.;  and  its  top  and 
side  views  equally  distant  from  the  G  L,  and  V  T  of  P,  respectively. 

Rules  1,  4,  5,  and  7  (Art.  74)  may  here  be  noted. 

(m)  Views  could  in  like  manner  be  obtained  upon  planes  auxiliary  to  the 
co-ordinate  or  principal  planes,  V,  H,  and  P;  namely,  upon  a  profile  plane  at  the 
left  of  the  object,  upon  a  bottom  plane  ||  to  H,  upon  a  rear  plane  ||  to  V,  and 
upon  planes  oblique  to  either  two  of  the  co-ordinate  planes. 

We  may  thus  have  front,  top,  right  and  left  side,  bottom,  rear,  and  oblique 
views. 

With  the  exception  of  the  latter,  the  relative  arrangement  of  these  views  in 
a  mechanical  drawing  would  generally  be  as  indicated  in  Fig.  120.  See  Art.  (i). 

Note  that  the  line  of  each  view  (excepting  the  rear)  nearest  the  front  view 
represents  the  front  line  or  surface  of  the  object  in  that  view. 


ORTHOGRAPHIC  PROJECTION 


49 


66.  To  draw  the  front,  top,  and  side  views  of  a  rectangular  object.  Suppose 
it  is  required  to  draw  these  views  of  the  block  (Fig.  119(a)),  its  position  and  dimen- 
sions being  given. 

(a)  First  draw  ONQ'  and  MNQ  to  represent  the  traces  of  V,  H,  and  P. 
That  portion  of  the  paper  below  O-N  and  to  the  left  of  M-N  will  represent  the 
front  plane,  that  above  O-N  and  to  left  of  N-Q  the  top  plane,  and  that  to  the 
right  of  M-N  and  below  N-Q'  the  right  side  plane.  The  outer  limiting  lines  of 
the  planes  are  always  omitted. 


FiG.II9(a) 


.Q1 


— 

Ev 

U  WIDTH 
BREADTH 

H 

cv 

V 


H-DEPTM^j 


t 


(b)  Since  the  distance  of  the  object  from  the  planes  is  immaterial,  the  views 
may  be  drawn  any  distance  from  the  G  L  and  traces  of  P.     Hence,  at  any  con- 
venient distance  below  O-N  and  to  the  left  of  M-N,  draw  rectangle  AVBVCVDV 
for  the  main  lines  of  the  front  view,   making   the  hor.  and  vert,   dimensions 
equal  to  the  breadth  and  height  respectively,  of  the  object.     The  opening  should 
at  first  be  disregarded. 

(c)  The  top  view  of  each  pt.  must  be  in  a  vert,  through  its  corresponding 
front  view,  Art.  65(1).      As  in  this  case  all  pts.  of  the  front  face  are  equally  distant 
from  V,  draw  the  projectors  from  the  front  view,  and  at  any  convenient  distance 
above  the  G  L,  draw  the  hor.  DH-CH  for  its  top  view. 

As  the  rear  face  is  ||  to  the  front,  its  top  view  will  be  EH-FH,  at  a  distance 
from  DH-CH  equal  to  the  depth  of  the  object.  As  the  side  faces  are  _L  to  H,  their 
top  views  will  be  the  verts.  DH-EH  and  CH-FH,  thus  completing  the  rectangle 
DHCHFHEH  for  the  main  lines  of  the  top  view  of  the  object. 

(d)  From  Art.  65(1)  it  follows  that  the  side  view  of  each  pt.  will  be  in  the 
hor.  through  its  corresponding  front  view  and  at  the  same  distance  from  the  V  T 
as  its  top  view  is  above  the  G  L.     Hence,  draw  indefinite  hors.  from  the  front 
view,  and  from  the  top  view  to  N-Q'. 

Then  as  N-Q  and  N-Q'  both  represent  the  H  T  of  P,  describe  arcs  with  N  as 
center,  to  carry  the  pts.  from  N-Q  to  N-Q'.  From  these  pts.  draw  verts,  to 
intersect  the  projectors  from  the  front  view,  thus  determining  Bp,  Gp,  Fp,  and 
Cp  and  forming  the  main  lines  of  the  side  view. 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


(e)  Similarly,  if  the  side  view  is  first  obtained,  the  reverse  of  the  above  process 
will  determine  the  top  view. 

(f)  Now,  returning  to  the  front  view,  draw  the  rectangle  to  represent  the 
opening,  then  project  as  before,  to  complete  the  other  views. 


LEFT  SIDE  VIEW 


TOP  VIEW 


]     FRONT  VIEW  J 


BOTTOM    VIEW 


RIGHT  SIDE  VIEW 


REAR  VIEW 


FlG.  1 2O 


(a) 


... 

- 

.- 

r 

h- 

— 

(b) 





—  j+F 

FIG.  121 

(g)  PRACTICAL  METHODS.  In  practical  drafting,  only  such  portions  of  the 
projectors  are  penciled  as  may  be  necessary  to  locate  the  required  pts.  The 
traces  of  the  planes  of  proj.  are  also  omitted;  center,  base,  or  other  lines  ||  to 
the  positions  of  the  traces  being  utilized  as  reference  lines  in  locating  the  views, 
practically  as  though  such  lines  were  the  actual  traces  of  the  planes.  Thus,  in 
determining  the  views  (Fig.  119(a))  the  lower  hor.  of  the  top  view  and  left  vert,  of 
the  side  view  could  have  been  located  any  convenient  distance  from  the  front 
view,  and  the  depth  set  off  from  top  to  side  or  vice  versa,  by  means  of  the  dividers. 

Fig.  121  (a)  shows  the  views  with  the  traces  and  projectors  omitted.  In 
this  case  all  measurements  were  set  off  with  reference  to  lines  representing  the 
axes  of  the  object,  called  center  lines. 


ORTHOGRAPHIC  PROJECTION 


51 


It  is  evident  that  a  C.  L.  of  a  view  may  be  regarded  as  the  trace  of  a  central 
plane  JL  to  the  plane  of  that  view. 

Fig.  121  (b)  shows  the  same  object  in  a  different  position.  Working  drawings 
of  an  object  composed  of  several  rectangular  parts  are  shown  in  Figs.  179,  180. 

67.  Objects  Having  Surfaces  Oblique  to  the  Co-ordinate  Planes.  The  form 
of  an  object  is  frequently  such  that  some  of  its  surfaces  and  lines  will  be  oblique 
to  the  planes  of  one  or  more  of  the  views;  and,  therefore,  foreshortened  in  those 
views.  Thus  in  the  pyramid  (Fig.  122)  the  front  and  rear  faces  are  oblique  to  V 
and  H,  the  left  and  right  faces  to  H  and  P,  and  its  slant  edges  to  V,  H,  and  P. 
These  surfaces  and  lines  are,  therefore,  foreshortened  in  all  of  the  views. 

The  base  is  1 1  to  H  and  has  two  of  its  edges  1 1  to  V  and  two  1 1  to  P.  When  thus 
placed  the  object  is  ||  to  V,  H,  and  P  as  much  as  it  can  be,  since  its  axes  or  principal 
dimensions  are  also  ||  to  those  planes,  and  the  views  enable  the  form  of  the  object 
as  a  whole  to  be  determined,  as  in  the  case  of  a  rectangular  object. 

Rules  2,  3  and  6  (Art.  74)  may  here  be  noted. 


FIG.  123 


A       rpr       D 
FIG.  124 


68.     Objects  Having  Curved  Surfaces. 

(a)  Any  view  of  a  sphere  is  a  O  equal  to  a  great  O  of  the  sphere,  Fig.  123. 
Observe  that  the  great  O  ABCD  is  ||  to  V,  AECF  H  to  H,  and  DEBF  ||  to  P. 

The  view  of  a  right  circular  cylinder  upon  a  plane  to  which  its  axis  isj_  is  a  O. 
Its  view  upon  a  plane  ||  to  the  axis  is  a  rectangle.  Thus  in  Fig.  124  the  bases 
are  ||  to  H  and  _L  to  V;  they  are,  therefore,  seen  upon  H  in  their  exact  shape, 
and  upon  V  as  ||  hors.  equal  to  the  diam.  of  the  base.  The  elements  are  ||  to  V 
and  J_  to  H  and,  therefore,  seen  upon  V  in  their  exact  length,  and  upon  H  as  pts. 
in  the  O . 

The  views  of  a  right  circular  cone  whose  axis  is  thus  related  to  the  planes  are 
aO  and  a  A.  Fig.  125.  The  outer  elements  only  being  ||  to  V,  they  only  are 
seen  upon  it  in  their  exact  length.  Intermediate  elements  are  unequally  inclined 
to  V  and,  therefore,  unequal  upon  it.  As  all  the  elements  are  equally  inclined 
to  H,  they  are  equally  foreshortened  upon  H..  Note  that  as  the  third  view  in  each 
case  would  be  the  same  as  one  of  the  others,  its  representation  is  unnecessary. 


H 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


FIG.  126 


(b)  Equidistant  elements  of  a  right 
circular  cylinder  or  cone  may  be  deter- 
mined by  equal  division  of  the  base  or 
other  O  of  the  surface,  in  the  view  in 
which  that  O  is  seen  in  its  exact  shape. 
Twelve  are  usually  sufficient. 

(c)  Fig.  120  represents  a  truncated 
cylinder.     As  the  elements  are  _L  to  H, 
the  top  view  of  the  curve  of  the  oblique 
surface  coincides  with  that  of  the  base. 
As  the  surface  is  oblique  to  P,  its  side 
view   will  also  be  a  curve.     To  draw 
this  view  it  is  necessary,  since  there  are 
no  vertices  as  in  rectilinear  figures,  to 
obtain  a  sufficient  number  of  its  deter- 
mining pts.,  by  assuming  these  first  in 
the  known  (front    and    top)   views  of 

in  the  side    view    the   curve   is   drawn 


the  curve.     Having  located  these  pts. 
through  them,  as  shown.     See  Art.  18(a). 

In  cylindric  or  conic  surfaces  these  pts.  may  be  regarded  as  the  ends  of  elements. 

69.  Projections  upon  Oblique  Auxiliary  Planes.  When  it  is  necessary  to 
represent  an  object  so  that  some  particular  surface,  _L  to  one  plane  but  oblique 
to  another,  shall  be  shown  in  its  true  or  exact  shape,  an  auxiliary  view  upon  a  plane 
||  to  that  surface  may  be  obtained. 

(a)  In  Fig.  127,  DABACA,  etc.,  represents  a  proj.  upon  an  aux.  plane  A, 
||  to  the  surface  DEC,  that  is,  upon  a  plane  JL  to  V,  but  oblique  to  H.  The 
method  of  obtaining  this  view  differs  from  that  of  obtaining  a  side  view  only  in 
that  the  V  T  in  this  case  is  ||  to  DVBVCV,  instead  of  _L  to  the  G  L.  Having 

located  the  traces  of  A, 
draw  _Ls  to  the  V  T  from 
the  front  view  and  inter- 
sect these  by  projecting 
from  the  top  view  as 
shown,  or  transfer  the  pts. 
with  dividers  as  in  Art. 
6G(g).  Joining  the  pts. 
thus  determined  will  form 
the  required  view.  Note 
application  of  principle  in 
Figs.  144,  182. 

(b)  In  Fig.  128(a)  an 
aux.  view  AABADA,  upon  a 
plane,  A,  _L  to  H  and  ||  to 
a  central  plane,  Y-Y,  is 
shown.  In  this  case  the 
FIG  127  view  was  determined  by 


ORTHOGRAPHIC  PROJECTION 


53 


means  of  C.  Ls.  and  base  lines  regarded  as  the  traces  of  planes  _L  to  those  of  the 
views.  The  method  differs  from  that  of  obtaining  the  front  view  in  Fig.  127  only 
in  the  changed  positions  of  the  central  planes  X-X  and  Y-Y.  Thus  Y-Y  is  the 
H  T  of  the  given  central  plane;  X-X  is  the  H  T,  and  trace  upon  A,  of  a  second 
central  plane  _L  to  Y-Y. 
Z-Z  is  the  V  T  of  the 
plane  of  the  base,  and 
Z'-Z'  drawn  any  con- 
venient distance  from 
Y-Y  and  ||  to  it  is  the 
trace  of  the  base  plane 
upon  A. 

Perpendiculars  to 
Z'-Z'  from  AH,  BH,  and 
CH  determine  AA,  BA, 
and  CA;  setting  off  the 
height  of  D  above  Z-Z 
from  the  front  view  and 
joining  DA,  AA,  and  BA 
completes  the  required 
view. 

(c)  Fig.    128  (a)    also 

method  of  obtaining  an  aux.  view 
DBCBBB,  upon  a  plane,  B,  ||  to  a  sur- 
face DBC,  which  is  not  _L  to  the  plane 
of  either  of  the  given  views,  that  is,  not 
shown  as  a  line,  but  _l_  to  which  a 
plane,  Y-Y,  can  be  directly  determined. 

The  aux.  view  AABADA,  is  first  ob- 
tained as  in  the  preceding  case.  Y'-Y' 
||  to  DABACA  is  the  trace  upon  B  of  the 
_]_  plane  Y-Y. 

Perpendiculars    to    Y'-Y'    from   A 
and  DA  determine  AB  and  DB;  setting 
off  the  distances  of  B  and  C  from  Y-Y, 
_L   to  Y'-Y',  and  joining  pts.  AB,  BB, 
CB,  DB  completes  the  required  view. 

(d)  Fig.  (b)  shows  the  method  of 
obtaining  an  aux.  view  upon  a  plane, 
B,  _L  to  the  axis  of  an  object  which  is 
oblique  to  V,  H,  and  P.     Note  that  the 

views  upon  A  and  B  determine  the  true  dimensions  of  the  prism. 

(e)  In  a  curvilinear  object,  it  is  necessary  to  assume  pts.  in  the  known  views 
of  the  curves  and  then  obtain  these  in  the  aux.  view,  as  is  evident  from  Art.  68 (c). 

(f)  When  the  true  shape  of  a  particular  surface  only  is  required  the  same 
method,  or  that  of  Art.  70,  may  be  used. 


M 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


70.  Revolution  of  Surfaces.  It 
is  frequently  desirable  to  determine 
the  true  shape  of  a  surface  by  revolv- 
ing it  about  an  axis  until  ||  to  V,  H, 
or  P. 

(a)  Fig.  129(a)  represents  a 
truncated  hexagonal  prism  with  its 
oblique  surface  ABCDEF  revolved 
about  an  imaginary  axis  _L  to  V 
through  pt.  A,  until  it  is  ||  to  H,  upon 
which  it  is,  therefore,  seen  in  its  true 
shape  A'B'C'D'E'F'. 

In  revolving  a  surf  ace,  each  pt.  not 
in  the  axis  of  revolution  describes  an 
arc  whose  plane  is  J_  to  that  axis. 
In  this  case,  therefore,  the  arcs  are 


(b) 


>>K       FIG.  129 


seen  as  arcs  in  the  front  view  and  as  lines  ||  to  the  G  L,  or  C.  L.,  AH-D',  in  the 
top  view.  Hence,  the  front  view  of  the  surface  remains  a  line  of  the  same  length, 
and  the  distances  of  the  pts.  of  the  surface  from  V  also  remain  unchanged. 

The  surface  could  in  like  manner  be  revolved  ||  to  H,  or  to  P,  about  any  other 
axis  JL  to  V.     Thus  A"B"C"D"E"F"  represents  it  revolved  about  1-2  until  1 1  to  P. 

Fig.  (b)  represents  the  surface  revolved  about  its  side  B-C  until  ||  to  V. 
As  B-C  is  ||  to  V,  the  front  views  of  the  arcs  described  by  A,  F,  E,  and  D  are  J_s 
to  B-C  at  Av,  Bv,  Cv,  and  Dv;  and  as  the  true  distances  of  A,  F,  E,  and  D  back 
of  B-C  are  seen  in  the  top  view,  these  distances  are  set  off  on  the  _Ls  at  A',  F', 
E',  and  D'  as  shown. 

The  figure  also  shows  a  lateral  surface  revolved  about  its  base  edge,  G-H, 
until  ||  to  H;  and,  about  the  lateral  edge,  A-G,  until  ||  to  V.  Note  application  of 
principle  in  Fig.  140  to  the  revolution  of  an  imaginary  surface  on  C-D.  See  also 
Figs.  145,  153,  158,  159,  161  (b). 


ORTHOGRAPHIC  PROJECTION 


55 


(b)  In  Fig.  128(a),  a  surface  DEC,  oblique  to  V,  H,  and  P,  is  shown  revolved 
||  to  H  about  its  base  edge  B-C.  As  B-C  is  ||  to  H,  the  top  view  of  the  arc 
described  by  pt.  D  will  be  a  _L  to  BH-CH,  coinciding  with  Y-Y.  Hence,  the 
true  distance  of  D  from  B-C  is  determined,  in  this  case,  by  an  aux.  view  upon 
a  plane  A,  ||  to  the  central  plane  Y-Y,  to  which  the  surface  DEC  is  JL .  Art.  69. 
Rule  8  (Art.  74)  may  here  be  noted. 

7 1 .     True  Length  and  Position  of  Lines  Oblique  to  the  Co-ordinate  Planes.      To 

determine  the  true  length  of  a  line  oblique  to  the  planes,  the  line  may  be  projected 
upon  a  ||  aux.  plane,  or  revolved  ||  to  a  plane  of  proj.,  practically  as  in  the  case 
of  a  surface. 

(a)  FIRST  METHOD.     To  obtain  the 
true  length  of  A-D  (Fig.  128(a))  upon  an 
aux.  plane  _L  to  H.     The  method  is  evi- 
dent from  Art.  69 (b).     AA-DA  is  the  true 
length,  and  the  Z  it  makes  with  the  H  T 
of  plane  Y-Y  containing  the  line  is  the  true 
Z  the  line  itself  makes  with  H. 

To  obtain  the  true  length  of  A-D  (Fig. 
130)  upon  an  aux.  plane  A  _|_  to  V,  regard 
Av  -Dv  as  the  V  T  of  the  plane  containing 
the  line;  Y-Y  as  the  H  T  of  a  central  plane 
_L  to  the  first;  and  Y'-Y'  placed  any 
distance  from  AV-DV  and  ||  to  it,  as  the 
trace  of  this  second  plane  upon  the  aux. 
plane  A. 

Projecting  from  Av  and  Dv  _J_  to  Y'-Y7 

obtain  DA,  and  set  off  the  _L  distance  that  A  is  back  of  Y-Y.  Then 
AA-DA  will  be  the  required  true  length  and  (a)  the  true  Z  that  the  line  itself 
makes  with  V. 

(b)  SECOND  METHOD.      If  the  line  A-D  be  revolved  ||  to  V  about  an  axis  _]_ 
to  H  through  D,  the  line  remaining  at   the   same   Z   with  H  and  pt.  D  fixed, 
pt.  A  will  describe  a  hor.  arc,  the  top  view  of  which  will  be  the  arc  AH-A',  and  the 
front,  the  hor.  AV-A'.     In  this  position,  therefore,  the  top  view  of  the  line  remains 
unchanged  in  length,  while  the  front  A'-DV  shows  the  true  length  of  A-D  and  the 
true  Z  (b)  that  it  makes  with  H. 

As  all  of  the  edges  meeting  in  the  vertex  D  are  equal  and  equally  inclined  to 
H,  A'-DV  is  the  true  length  and  (b)  the  true  Z  of  all. 

Again,  as  all  pts.  in  the  revolved  line  remain  on  the  same  levels,  they  describe 
similar  hor.  arcs.  The  true  distance  from  D  of  any  pt.,  as  E  on  D-C,  may,  there- 
fore, be  found  by  drawing  a  hor.  from  that  pt.  to  the  true  length  line  A'-DV,  as 
proved  by  the  top  view  which  shows  E  revolved  into  A'-DH.  E'-DV  is,  there- 
fore, the  true  length  of  E-D. 

The  figure  also  shows  the  line  A-D  revolved  1 1  to  H  about  an  axis  _L  to  V, 
through  D.  A"-DH  is,  therefore,  the  true  length  and  (c)  the  true  Z  of  A-D  with  V. 

The  line  could  in  like  manner  be  revolved  about  an  axis  through  A. 


FIG  130 


M 


ESSENTIALS  OF  MECHANICAL  DRA1  I  [NQ 


,. 


-Tb 

H 

-4 

Y'l 

\ 

I 

I 

I 
1 

| 

1 
1 
1 

FIG.  131 


72.  Objects  Ob- 
lique to  the  Co-ordi- 
nate Planes.  It  fre- 
quently occurs  that 
the  axes  of  one  or 
more  integral  parts  of 
an  object  are  oblique 
to  those  of  the  main 
portion,  and,  there- 
fore, to  the  planes  of 
one  or  more  of  the 
views.  Thus,  in  Fig. 
193  the  left  and  right 
side  pieces  of  the  taboret  are  oblique  to  V  and  P; 
in  Fig.  194  the  braces  (A)  are  oblique  to  V  and  H, 
and  braces  (B)  to  H  and  P  (see  also  bolts  in  Fig. 
182) ;  in  Fig.  199  the  legs  are  oblique  to  V,  H,  and  P. 
When  the  detail  or  part  is  ||  to  V,  H,  or  P 
only,  its  view  upon  that  plane  should,  as  a  rule, 
first  be  determined,  since  that  view  only  will 
show  two  of  its  dimensions  in  their  exact  size. 

When  not  ||  to  either  V,  H,  or  P,  it  may,  if 
necessary,  be  represented  first  as  ||  to  one  of  those 
planes  and  then  as  revolved  to  the  required  posi- 
tion; or,  the  method  of  Art.  73  may  be  used. 

(a)  In  Fig.  131,  (a)  represents  a  rectangular 
prism    ||   to   V,    H,   and   P.     (b)  represents   this 
prism  ||  to  H  but  oblique  to  V  and  P,   that  is, 
turned  backward  at  the  left  from  the  position  of 
(a),  through  an  Z   of  30°  about  an  axis  _L  to  H. 
(c)  represents  it  1 1  to  V  but  oblique  to  H  and  P, 
that  is,  turned  to  the  right  from  the  position  of 
(a),  30°  about  an  axis  JL  to  V.      (d)  represents  it 
||  to  P  but  oblique  to  H  and  V,  that  is,  turned  for- 
ward from  the  position  of  (a),  30°  about  an  axis 
_L  to  P. 

(b)  By  experiment  with  the  actual  object,  it 
will  be  observed  that  the  positions  of  its  lines  rela- 
tive to  the  plane  to  which  the  axis  of  revolution  is 
_L  remain  unchanged.     It  follows  that  the  object 
without  changing  the  form  and  dimensions  of  the 


may  be  turned  through  any  Z 
view  upon  that  plane.  Thus,  in  Fig.  131,  the  top  view  in  (b),  the  front  view  in 
(c),  and  the  side  view  in  (d)  are  the  same  as  the  corresponding  views  in  (a), 
save  that  they  show  the  changed  relation  of  the  object  to  the  planes  to  which 
the  axis  of  revolution  is  ||.  The  views  upon  the  latter  planes  change  in  outline 
with  each  position  of  the  revolution,  but  as  no  pt.  changes  its  distance  from  the 


ORTHOGRAPHIC  PROJECTION 


57 


(a) 


plane  to  which  the  axis  is  _l_ ,  the  dimen- 
sions [|  to  the  axis  also  remain  un- 
changed. Thus,  corresponding  pts.  of 
the  front  and  side  views  in  (b)  and  (a) 
are  at  the  same  distances  from  the  G  L, 
and  H  T  of  P;  those  of  the  top  and  side 
views  in  (c)  and  (a)  at  the  same  distances 
from  the  G  L,  and  V  T  of  P;  and  those 
of  the  front  and  top  views  in  (d)  and  (a) 
at  the  same  distances  from  the  traces  of 
P.  Note  Rule  8,  Art.  74. 

(c)  In  determining  the  views  of  an 
object   thus   revolved,   having    located 
the  G  L  and  traces  of  P,  or  equivalent 
reference   lines*,   draw   the    main   axis 
or  axes  of  the  unchanged  view  at  the 
required  Zs  with  the  planes  to  which 
the  object  is  oblique.     (See  lines  A-A 
andB-Bin  Fig.  (b).)     Measuring  upon, 
and  1 1  to  these  axes,  construct  the  view. 
From    this  view  draw  projectors  to    a 
second  plane  and  set  off  upon  them  the 
unchanged  dimensions  ||  to  the  axis  of 
revolution,  measuring  from  the  trace  of 
the  first  plane,  or  equivalent  reference 
line.     Joining  the  pts.  thus  determined 
will  complete  the  second  view.      From 
these,  the  third  may  be  obtained  as  in 
Art.  66. 

(d)  In  Fig.  132,  (a)  and  (b)  repre- 
sent the  same  prism  when  oblique  to  V, 

H,  and  P.  (a)  represents  it  as  turned  to  the  right  from  the  position  of  (b)  in 
Fig.  131,  30°  about  an  axis  J_  to  V;  (b)  as  turned  backward  from  the  position 
of  (a),  30°  about  an  axis  J_  to  P. 

In  determining  these  views  the  same  principles  apply  as  in  the  preceding  cases. 
By  assuming  and  representing  an  object  in  a  primary  position,  as  nearly  as  may 
be  like  that  required,  and  then  as  revolved,  any  conceivable  position  may  be 
represented. 

Observe  that  the  side  view  in  Fig.  (b)  was  plotted  from  Fig.  (a)  by  Art.  57. 

(e)  In    an    object    having    curved  surfaces,   pts.   must  be  assumed  in  the 
primary  positions  of  the  curves,   as  explained  in  Art.  68 (c).     Note  applications 
of  principle  in  Fig.  193. 


*In  practical  work  the  traces  and  projectors  are  omitted,  as  explained  in  Art.  66(g).     They  are 
shown  in  Fig.  131  merely  for  illustration. 


58  KSSENTIALS  OF  MECHANICAL  DRAFTING 

(f)  Instead  of  representing  each  position  separately,  the  required  views  may 
be  obtained  by  revolution  about  a  C.  L.  of  a  primary  position  as  in  Fig.  133(a), 
or  about  an  axis  ||  to  that  C.  L.  as  in  Fig.  (b). 

73.     Partial  Views,  and  Use  of  Auxiliary  Views  in  Determining  Required  Views. 

(a)  Partial  views,  and  aux.  views  (views  of  revolved  lines  or  surfaces,  and 
oblique  views)  may  frequently  be  used  in  place  of  complete  or  regular  views 
and  thus  economize   space  and  labor.     Thus  in  Fig.  214  half  of  the  top  view 
would  suffice  to  determine  the  required  pts.  of  the  front;  in  Fig.  153  a  revolved 
half-base  is  used  for  the  same  purpose;  in  Fig.  147  half  only  of  the  side  view 
of  A  and  C  is  necessary;   in  Fig.  144  half  of  the  oblique  view  would  suffice;  in 
Fig.   145  the  revolved  half-bases  of  (B)  and  (C)  dispensed  with  a  side  view;  in 
Fig.  146  a  view  _L  to  the  axis  of  (B)  or  a  revolved  imaginary  O  is  necessary  to 
determine  positions  of  elements. 

(b)  In  representing  objects  oblique  to  the  co-ordinate  planes,  primary  po- 
sitions may  be  omitted  and  the  missing   dimensions  of  the  required  views  deter- 
mined by  the  same  general  methods. 


FIG.  133 


To  illustrate,  Fig.  134(b)  represents  a  pentagonal  prism  with  its  axis  and 
lateral  edges  ||  to  H  and  at  30°  back  to  the  right  with  V.  Its  bottom  face  also 
is  ||  to  H.  Instead  of  representing  the  prism  first  as  ||  to  V  and  H  as  in  Fig.  (a), 
and  then  as  revolved  about  an  axis  _L  to  H  until  its  lateral  edges  make  the  required 
L  of  30°  with  V,  the  representation  of  the  first  position  may  be  omitted  and 
the  missing  dimensions  determined  as  follows: — 

First  locate  X-X,  Y-Y,  Z-Z,  and  Y'-Y'  for  the  C.Ls.  of  the  required  views. 
Through  the  center  of  the  top  view  draw  V-V  at  30°  with  Y-Y  for  the  H  T  of  a 
central  plane  containing  the  axis  and  W-W  for  the  H  T  of  a  central  plane  _L  to 
V-V.  Equally  distant  from  the  center  and  at  a  distance  apart  equal  to  the  axis 
of  the  prism,  draw  indefinite  _Ls  to  V-V  for  the  bases.  Now  obtain  an  aux.  view 
upon  a  plane  ||  to  the  central  plane  W-W.  The  H  T  of  this  plane  would  be  || 


ORTHOGRAPHIC  PROJECTION 


59 


to  W-W,  and  the  trace  of  the  hor.  central  plane  Z-Z,  upon  the  aux.  plane,  would 
also  be  ||  to  W-W.  Hence,  at  any  convenient  distance  from  either  base,  draw 
this  trace  Z'-Z'  and  about  its  intersection  with  V-V  construct  the  view,  noting 
that  as  the  lower  surface  of  the  prism  is  ||  to  H,  the  line  A-B  will  be  ||  to  Z'-Z', 
as  shown.  Projecting  perpendicularly  to  W-W  from  this  view,  complete  the  top 


view.  Project  next  to  V  and  complete  the  front  view,  obtaining  the  distances  of 
pts.  above  and  below  Z-Z  by  measuring  from  Z'-Z'  in  the  aux.  view.  The  missing 
dimension  in  this  case  could  also  be  obtained  by  revolving  a  base  ||  to  H,  then  by 
counter  revolution  projecting  back  to  the  top  view,  as  shown. 

In  some  cases  a  view  upon  an  aux.  plane  ||  to  the  central  plane  V-V,  as 
shown,  would  be  necessary  instead  of,  or  in  addition  to,  that  of  the  end. 


60  ESSENTIALS  OF  MECHANICAL  DRAFTING 

Fig.  135  represents  the  same  prism  with  its  axis  ||  to  V  and  at  60°  up  to  the 
right  with  H.  Its  rear  face  also  is  ||  to  V. 

Fig.  136  represents  it  with  its  axis  ||  to  P  and  30°  forward  with  V.  Its  left 
face  also  is  ||  to  P. 


The  methods  of  obtaining  the  missing  dimensions  are  identical  with  those  of 
the  preceding  case  and  are  shown  by  the  figures. 

(c)     Fig.   137  represents  the  prism  with  its  axis  inclined  up  to  right  at  60° 
with  H  and  in  a  plane  at  45°  back  to  left  with  V.     One  of  rear  faces  also  is  at  45° 


ORTHOGRAPHIC  PROJECTION 


61 


FIG.  137 


with  V.  This  is  identical  with 
a  revolution  from  the  position 
of  Fig.  135,  45°  about  a  vert, 
axis,  but  instead  of  represent- 
ing the  prism  first  at  60°  with 
H,  as  in  Fig.  135,  the  work  in 
this  and  similar  cases  may  be 
shortened  as  follows: — 

First  locate  the  C.Ls.  X-X, 
Y-Y,  and  Z-Z  for  the  required 
views.  Through  the  center  of 
top  view  draw  U-U  at  45°  for 
the  H  T  of  a  central  plane  con- 
taining the  axis,  and  V-V  _L  to 
U-U  for  the  trace  of  a  central 
plane  _L  to  H  and  to  an  aux. 
plane  A  1 1  to  the  axis.  Through 
any  convenient  point  on  V-V 
draw  W-W  at  60°  with  U-U  as 
the  C.L.  of  the  aux.  view,  and 
about  the  intersection  of  V-V 

and  W-W  construct  the  aux.  view  as  in  Fig.  135.  From  these  aux.  views  obtain  the 
top  view.  Observe  that  this  is  identical  with  revolving  both  front  and  top  views 
of  Fig.  135,  through  an  Z  of  45°.  Project  next  from  H  to  V,  and  obtain  the  dis- 
tances of  pts.  above  and  below  Z-Z  by  measuring  from  Z'-Z',  regarded  as  the  trace 
of  the  hor.  central  plane  Z-Z  upon  the  aux.  plane.  Connect  the  proper  pts.  to 
complete  the  front  view. 

74.    Rules  Governing  the  Position  of  Lines  and  Surfaces  Relative  to  Any  Two 
Perpendicular  Planes  of  Projection. 

(1)  A  line  J_  to  a  plane  is  ||  to  the  _|_  plane.      Its  view  upon  the  first  is  a  pt.;  its  view  upon  the 
second  is  a  line  ||  and  equal  to  the  line  itself  and  J_  to  the  trace  of  those  planes. 

(2)  A  line  oblique  to  one  plane  and  |  [  to  a  J_  plane  has  its  view  upon  the  first  a  line  shorter 
than  the  line  itself  and  ||  to  the  trace  of  those  planes;  the  Z  the  other  view  makes  with  the  trace  is 
equal  to  the  Z  the  line  itself  makes  with  the  first  plane. 

(3)  A  line  oblique  to  two  J_  planes  has  its  views  upon  both  shorter  than  the  line  itself  and  neither 
view  shows  the  true  length,  nor  true  Zs  the  line  itself  makes  with  those  planes. 

(4)  The  views  of  1 1  lines  are  1 1  unless  their  views  coincide,  or  are  merely  pts. 

(5)  A  surface  1 1  to  a  plane  is  _1_  to  the  _|_  plane.     Its  view  upon  the  first  plane  is  a  figure  similar 
and  equal  to  the  surface  itself.     Its  other  view  is  a  line  1 1  to  the  trace  of  the  planes. 

(6)  A  surface  oblique  to  one  plane  and  J_  to  the  _L  plane  is  foreshortened  1 1  to  the  trace  in  its 
view  upon  the  first.     The  Z  the  other  view  makes  with  the  trace  of  those  planes  is  equal  to  the  Z 
the  surface  itself  makes  with  the  first  plane. 

(7)  The  views  of  ||  and  equal  surfaces  whose  corresponding  lines  are  ||  are  similar  figures  whose 
corresponding  lines  are  equal  and  ||,  unless  the  views  coincide,  or  are  merely  lines. 

(8)  If  a  line  or  figure  be  revolved  from  one  position  to  another  about  an  axis  _L  to  a  plane,  its 
view  upon  that  plane  remains  unchanged  in  form  and  dimensions.     In  the  view  upon  a  J_  plane  the 
distances  of  the  pts.  from  the  trace  of  the  planes  also  remain  unchanged. 


CHAPTER  VI 
PLANE    SECTIONS 

75.  Principles  and  Methods.     If  a  plane  be  imagined  to  intersect  an  object, 
the  portion  of  the  object  lying  within  it  is  called  a  plane  section,  and  the  plane 
a  cutting  or  section  plane,  denoted  by  C.  P. 

(a)  If  the  portion  of  the  object  hiding  the  section  be  imagined  as  removed, 
the  view  of  the  remaining  portion  or  merely  that  of  the  section  itself,  upon  a 
plane  ||  to  the  C.  P.,  is  called  a  sectional  view.      A  sectional  view  thus  shows 
the  sec.  in  its  true  shape.     Such  views  are  made  in  place  of  or  in  addition  to 
external  views  to  show  the  form,  dimension,  and  arrangement  of  hidden  parts 
more  clearly;  to  aid  in  determining  the  views  of  irregular,  oblique,  and  inter- 
secting forms;  and  to  enable  the  draftsman  to  dispense  with  numerous  dotted  lines. 

The  figures  of  this  chapter  show  sees,  of  geometric  forms;  practical  applica- 
tions are  shown  in  Chap.  XL 

(b)  When  a  sec.  is  not  ||  to  V,  H,  or  P,  its  true  shape  may  be  found    by 
projecting  upon  an  aux.  plane  ||  to  the  C.  P.,  as  in  Art.  69;  or  by  revolving  the 
sec.  ||  to  V,  H,  or  P,  as  in  Art.  70. 

(c)  The  position  of  a  C.  P.  is  indicated  in  the  view  in  which  it  is  seen  edge- 
wise; that  is,  by  its  trace  upon  V,  H,  or  P. 

The  section  is  generally  indicated  by  drawing  ||s  across  it  called  section 
lines,  usually  at  45°  up  to  the  right.  The  spacing  is  dependent  upon  the  size 
and  shape  of  the  sec.,  usually  yV",  judged  by  eye. 

Avoid  drawing  the  lines  ||  to  the  main  lines  of  a  sec. 

When  a  drawing  is  to  be  inked,  it  is  not  desirable  to  pencil  the  sec.  lines, 
but  the  sectioned  portion  may  be  indicated  by  a  few  lines  sketched  freehand. 
In  views  which  show  the  true  shape  only,  sec.  lines  may,  for  economy  of  time, 
be  omitted. 

(d)  In  obtaining  a  sectional  view  the  trace  of  the  C.  P.  is  first  drawn.     This 
determines  the  pts.  in  which  the  C.  P.  cuts  the  edges,  elements,  or  other  lines 
of  the  object.     Next  project  these  pts.  to  the  corresponding  lines  in  the  other 
views,  and  join  them.     Then  proceed  as  in  (b). 

76.  Objects  Having  Plane  Surfaces.     As  the  intersection  of  plane  surfaces 
is  a  st.  line,  it  follows  that  any  plane  sec.  of  an  object  bounded  by  plane  surfaces, 
as  a  prism  or  pyramid,  will  be  a  polygon  of  3,  4,  or  more  sides  according  as  the 
plane  cuts  3,  4,  or  more  surfaces. 

(a)  A  PRISM.  Fig.  138(a)  represents  a  right  square  prism  with  its  lateral 
faces  at  45°  with  V,  cut  by  a  plane  A-B,  _1_  to  its  base  and  ||  to  V.  The  pts. 
of  intersection  1,  2,  3,  and  4  with  the  top  and  bottom  edges  are,  therefore, 
determined  in  the  top  view  and  then  projected  to  the  front.  As  the  plane  cuts 
four  _1_  faces,  the  sec.  is  a  rectangle;  and,  being  ||  to  V,  is  shown  in  its  true  shape 
in  the  front  view. 

62 


PLANE  SECTIONS  63 

Fig.  (b)  represents  the  prism  in  the  same  position,  cut  by  a  plane  _L  to  V, 
but  oblique  to  H  and  P.  The  pts.  of  intersection  are,  therefore,  determined 
in  the  front  view. 

Fig.  (c)  represents  the  prism  with  its  vert,  faces  at  30°  and  60°  with  V,  cut 
by  a  plane  _L  to  P  and  oblique  to  V  and  H.  This  illustrates  a  case  in  which 
the  sec.  is  not  symmetrical  with  respect  to  a  C.  L. 


FIG.  138 


(b)  A  PYRAMID.  In  Fig.  139  the  C.  P.,  D-E,  cuts  the  lateral  edges  of  the 
pyramid  at  1,  2,  3,  and  4.  Pts.  1  and  3  may  be  projected  to  the  top  view  in 
the  usual  manner;  pts.  2  and  4,  however,  are  in  edges  which  are  represented 
by  vert,  lines  in  both  views.  To  obtain  these  pts.  in  the  top  view  either 
of  the  following  methods  may  be  used. 

First  Method.  Draw  a  side  view  and  transfer  the  distance  2P-4P  from  it  to 
the  top  view. 


(it 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


Second  Method.  Imagine  the  edge  A-B  to  be  revolved  ||  to  V  to  coincide 
with  A-C.  Then  the  front  view  of  2  will  be  at  2'  and  its  top  view  at  2".  If 
now  the  line  A-B  be  revolved  back  to  its  original  position,  2H  will  be  determined 
as  shown;  4H  may  in  like  manner  be  found. 

TheC.  P.,  F-G,  gives  a  triangular  sec.  The  method  of  finding  pt.  7V  is  evident 
from  the  preceding. 

77.     Objects  Having  Curved  Surfaces. 

(a)  A  SPHERE.  Any  sec.  of  a  sphere  is  a  O  whose  center  is  in  the  diam. 
of  the  sphere,  ±  to  the  C.  P.  See  Art.  2(n). 

In  Fig.  140  the  top  view  of  sec.  on  A-B  is  thus  a  0  whose  diam.  is  equal 
to  1V-2V.  The  front  view  of  sec.  on  C-D  is  an  ellipse.  The  intersection  of  the 


FIG.  139 


FIG.  140 


C.  P.  with  the  hor.  great  O  of  the  sphere  determines  3  and  4  in  the  top  view. 
As  the  front  view  of  that  O  is  in  the  hor.  E-F,  these  pts.  will  be  projected 
at  3V  and  4V,  as  .shown.  Intermediate  pts.  of  the  curve  may  be  found  by 
determining  pts.  of  intersection  of  the  C.  P.  with  other  Os  of  the  sphere  whose 
planes  are  ||  to  either  H  or  V.  Thus,  assuming  a  O  parallel  to  V,  cutting  the 
O  of  the  sec.  at  5  and  6,  these  points  will  be  determined  in  the  top  view  and 
projected  at  5V  and  6V  in  the  front.  Again,  assuming  a  hor.  O  to  cut  the  sec. 
at  5  and  8,  these  pts.  will  be  determined  in  the  top  view,  and  projected  at  5V 
and  8V.  Note  that  the  highest  pt.,  7,  of  the  curve  is  that  in  which  a  hor.  O 
becomes  tangent  to  the  plane  of  the  sec. 

The  assumed  Os  may  be  regarded  as  lines  of  intersection  given  by  aux.  C.  Ps., 
each  of  which  cuts  the  plane  of  the  sec.  in  a  chord  of  both  Os,  as  5-6,  whose 
end  pts.  are  thus  in  the  required  curve. 


PLANE  SECTIONS 


65 


The  intermediate  pts.  of  sec.  C-D  may  also  be  found  by  the  reverse  of  the 
method  of  Art.  70.  Thus  the  half  sec.  revolved  ||  to  H  about  diam.  3-4 
determines  the  distance  of  pts.  5  and  6  from  diam.  3-4,  to  be  set  off  above  and 
below  that  line  in  the  front  view. 

(b)  A  CYLINDER.      Any  sec.   ||  to  the  axis  of  a  right  circular  cylinder  is  a 
rectangle,  and  a  sec.  ||  to  the  base,  a  O-     See  Art.  2(1). 

The  sec.  on  A-B  (Fig.  141)  is  an  ellipse  whose  top  view  is  a  O  coinciding  with 
the  view  of  the  base.  The  sectional  view  may  be  obtained  as  in  Art.  69 (e), 
or  by  determining  pts.  of  intersection  of  the  C.  P.  with  elements,  as  indicated. 

The  assumed  elements  may  be  regarded  as  lines  of  intersection  given  by  aux. 
C.  Ps.  which  cut  the  plane  of  the  sec.  in  lines,  as  2-6  and  3-5,  whose  ends  are  thus 
in  the  required  curve. 

(c)  A  CONE.     A  sec.  through  the  vertex  and  base  of  a  right  circular  cone  is 
a  A;    a  sec.  _L  to  the  axis,  a  O    whose  center  is  in  the  axis  of  the  cone.     See 
Art.  2(m). 

The  oblique  sec.  (Fig.  142)  is  an  ellipse,  pts.  for  the  top  and  sectional  views 
of  which  may  be  obtained  by  determining  the  pts.  of  intersection  with  elements 
of  the  cone.  A  more  accurate  method  is  to  assume  a  number  of  Os  of  the  cone 
to  cut  the  sec.  These  will  be  ||  hors.  in  the  front  view  and  Os  concentric  with 
the  O  of  the  base  in  the  top.  Thus,  drawing  the  front  view  of  a  O  through 
any  pt.,  as  lv,  3V,  and  projecting  from  lv,  3V,  to  the  top  view  of  the  O,  pts. 
1H  and  3H  will  be  determined. 

The  vert.  sec.  being  J_  to  both  V  and  H  has  both  front  and  top  views  a  st. 
line,  but  being  ||  to  P  will  show  its  true  shape,  a  hyperbola,  on  that  plane. 

The  C.  P.  cuts  the  right  element  at  2  and  the  O  of  the  base  at  4  and  5,  which 
pts.  may  be  obtained  in  the  sectional  view  in  the  usual  manner.  Intermediate 
pts.  may  be  obtained  by  use  of  elements,  but  it  is  more  accurate  to  assume  other 
Gs  of  the  cone,  to  intersect  the  sec.  Thus  a  O  cutting  the  sec.  at  6  and  7 
determines  the  distances  of  those  pts.  from  the  C.  L.  in  the  top  view. 

The  assumed  elements  and  Os  may  be  regarded  as  lines  given  by  aux.  C.  Ps., 
as  in  the  case  of  the  cylinder,  Art.  (b). 


CHAPTER    VII 


INTERSECTION    OF    SURFACES 


FIG   143 


78.  Principles  and  Methods.  The  line  or  lines  in  which  the  surface  of  one 
integral  part  of  an  object  intersects  that  of  another  part  is  called  a  line  of  inter- 
section. This  line  joins  the  pts.  in  which  the  edges,  elements,  or  other  lines  of 
each  part  meet  the  surface  of  the  other,  as  is  evident  from  Figs.  144(a),  145(a). 
The  problem,  therefore,  in  determining  the  line  of  intersection  is  to  find  the 
pts.  of  intersection  of  the  edges,  or  of  a  sufficient  number  of  other  lines,  with 
the  surfaces,  and  to  draw  the  line  through  the  pts.  thus  found. 

(a)  The  pt.  of  intersection  of  a  line  and  a  surface  is  determined  in  the  view 
in  which  the  surface  is  seen  edgewise  or  as  a  line, and  then  projected  to  the  other 
view  of  the  line  precisely  as  in  finding  the  pts.  of  a  plane  section. 

Thus,  in  Fig.  143,  line  A-B  intersects  the  front  face  of  the  object  at  1,  as 
determined  in  the  side  view,  for  if  it  be  assumed  to  intersect  the  left  side  face, 

pt.  C  would  be  seen  in  the  side  view  to  come 
above  the  object.  A-B  also  intersects  the 
right  side  face  at  2,  as  determined  in  the 
front  view. 

When  the  intersected  surface  is  not  shown 
as  a  line  in  either  of  the  required  views,  an 
aux.  view  which  will  so   represent   it    may 
sometimes  be  used.      The  pt.  of  intersection 
of  any  line  and  surface  may  be  determined 
by  passing  a  plane  through  the  line,   _L  to 
the  plane  of  either  of  its  views.       This  C.  P. 
cuts  the  given  surface  in  a  line  in  which  the 
required  pt.  of  intersection  must  lie,  for  both 
lines  are  in  the  same  plane. 
Thus,  in  Fig.  143,  a  plane  _L  to  H  and  P  gives  a  sec.  345  6,  and  as  the  lines 
of  the  sec.  are  in  the  surface  of  the  pyramid,  pts.  1  and  2,  in  which  A-B  intersects 
them,  must  be  the  required  pts.     Similarly  a  plane  JL  to  V,  gives  sec.  7  8  9  10,  by 
which  1  and  2  are  determined  in  the  top,  or  side  view. 

(b)  It  is  evident  that  a  plane  which  cuts  both  intersecting  parts  of  an  object 
would  give  a  plane  sec.  of  each,  and  the  pts.,  if  any,  in  which  the  lines  of  these 
sees,  intersect,  being  in  the  surfaces  of  both  parts,  must  be  pts.  in  the  required 
line  of  intersection.     By  assuming  a  series  of  such  C.  Ps.,  any  desired  number  of 
pts.  may  be  found. 

The  position  of  the  C.  P.  should,  when  possible,  be  so  chosen  that  the  sec. 
of  each  part  is  one  readily  obtained. 

(c)  In  problems  involving  intersections,  first  draw  each  part  as  complete  as 
possible  in  itself,  that  is,  without  regard  to  its  intersection.     Next  obtain  the 


INTERSECTION  OF  SURFACES 


67 


pts.  of  intersection  of  the  edges  or  elements  which  can  be  directly  determined 
in  either  of  the  views;  then  those  whose  pts.  must  be  found  by  means  of  C.Ps., 
or  aux.  views.  In  finishing,  only  such  lines  should  be  rendered  as  edges  or 
outlines  as  represent  the  edges  or  outlines  of  the  object  as  a  whole. 

79.  Objects  Having  Plane  Surfaces.*  Fig.  144  represents  the  intersection 
of  a  triangular  prism  (A)  and  an  oblique  hexagonal  prism  (B).  The  lateral 
edges,  A-B,  C-D,  E-F,  etc.,  of  (B)  intersect  the  left  vert,  surface  of  (A)  at  1, 
3,  5,  7,  9,  11.  As  this  surface  is  seen  as  a  line  upon  both  V  and  H,  the  pts.  are 
determined  in  both  views,  and  since  all  the  pts.  lie  in  the  same  plane  as  that 
of  the  surface,  the  line  of  intersection  is  simply  the  outline  of  a  plane  sec.  of 
(B).  In  this  case  it  is  an  irregular  hexagon  which  would  be  seen  in  a  side  view. 


FIG.  144 


(a) 


The  edge  A-B  intersects  the  upper  surface  of  (A)  also  at  2.  As  the  surface 
is  seen  as  a  surface  upon  H  and  as  a  line  upon  V,  pt.  2  is  determined  in  the  front 
view  and  then  projected  to  the  top.  Edges  C-D,  E-F,  I-J,  and  K-L  intersect 
the  front  and  rear  surfaces  of  (A)  also,  at  4,  6,  10,  and  12,  respectively.  As 
these  surfaces  are  seen  as  lines  upon  H,  the  pts.  are  there  determined  and  then 
projected  to  the  front  view.  Edge  G-H  and  the  right  vert,  edge  of  (A)  are  in 
the  same  plane  ||  to  V  and  thus  seen  to  intersect  in  the  front  view  at  8. 

Points  13  and  14  in  which  the  hor.  edges  of  (A)  intersect  the  surfaces  of  (B) 
are  not  directly  determined  in  either  view,  for  neither  of  the  intersected  surfaces 
is  there  seen  as  a  line.  By  obtaining  an  aux.  view  upon  a  plane  _L  to  the  axis 
of  the  oblique  prism,  the  intersected  surfaces  of  the  latter  will  be  seen  as  lines 
DA-BA  and  BA-LA  and  the  pts.  of  intersection  of  the  edges  of  (A)  determined 
at  13A  and  14A  as  shown.  These  pts.  may  then  be  projected  to  the  front  view. 
It  is  obviously  unnecessary  to  draw  the  complete  aux.  view. 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


When  the  preceding  method  is  not  convenient  or  possible,  the  pts.  may  be 
found  by  means  of  a  C.  P.;  thus,  if  the  plane  of  the  hor.  surface  containing  the 
edges  be  assumed  to  intersect  (B)  it  would  give  a  sec.,  the  intersections  of  whose 
sides  with  the  edges  of  the  hor.  surface  determine  13  and  14  in  the  top  view 
as  shown.  Again,  if  the  plane  of  the  front  surface  of  (A)  be  assumed  to  intersect 
(B)  it  will  give  a  sec.,  the  intersection  of  a  side  of  which  with  the  front  hor.  edge 


'  7 


of  (A)  determines  pt.  13  in  the  front  view.  Pt.  14  could  in  like  manner  be  found. 
It  is  evident  that  merely  those  lines  or  portions  of  the  sees,  which  determine 
the  required  pts.  need  be  drawn. 


80.     Objects  Having  Curved  Surfaces. 


FIG.  146 


(a)  Fig.  145  represents  the  inter- 
section of  two  circular  cylinders  (A) 
and  (B)  and  an  equilateral  triangular 
prism  (C).  The  line  of  intersection 
of  (A)  and  (B)  is  determined  by  the 
pts.  of  intersection  of  the  elements 
of  one  cylinder  with  the  surface  of 
the  other.  Thus  B,  D,  F,  and  H 
in  which  the  upper,  front,  lower,  and 
rear  elements  of  (B)  intersect  (A)  are 
determined  in  the  top  view,  for  the 
cylindric  surface  of  (A)  is  there  seen 
as  a  line  (circle). 

Intermediate  pts.  may  be  found 
by  assuming  intermediate  elements 
of  (A)  in  a  side  view,  or  of  (B)  as 
shown;  the  positions  of  the  elements 


INTERSECTION   OF   SURFACES  69 

of  the  latter  being  transferred  from  top  to  front  view  by  means  of  the  revolved 
half-base,  Art.  73. 

Note  also  that  C.  Ps.  ||  to  the  axes  of  the  cylinders  would  cut  elements  of 
both,  which  intersect  in  pts.  of  the  required  curve. 

The  views  of  prism  (C)  may  be  determined  by  means  of  a  side  view,  or  a 
revolved  half -base  as  shown.  The  pts.  of  intersection  of  the  hor.  edges  with 
cylinder  (A)  are  determined  in  the  top  view.  Note  that  the  rear  surface  inter- 
sects the  cylinder  in  an  element.  The  inclined  surfaces  intersect  (A)  in  elliptic 
curves,  intermediate  points  in  which  may  be  found  by  assuming  elements  of 
the  cylinder  and  determining  their  intersection  with  (C)  in  a  side  view;  or  by 
assuming  lines  upon  the  prism  ||  to  its  axis.  Thus  a  hor.,  5-6,  intersects  the 
cylinder  at  5,  determined  in  the  top  view. 


Again,  a  C.  P.  ||  to  the  axes  of  (A)  and  (C)  would  give  an  element  of  (A) 
and  two  lines  of  (C)  whose  pts.  of  intersection  must  be  in  the  required  curves 
as  shown.  Note  that  pts.  9  and  11,  in  which  the  direction  of  curvature  changes, 
are  at  the  intersection  of  lines  9-10  and  11-12  with  the  right  element  of  (A). 

(b)  In  determining  the  curve  of  intersection  of  cylinders  (A)  and  (B)  in  Fig. 
146,  the  same  methods  would  be  used  as  in  (A)  and  (B)  of  Fig.  145. 

Equidistant  elements  of  (B)  may  be  determined  by  an  oblique  aux.  view,  or  a 
revolved  half-sec,  taken  J_  to  the  axis,  as  shown. 

(c)  In  determining  the  intersections  in  Fig.   147,  the  C.   Ps.   may  be  taken 
horizontally,  or  through  elements,  as  shown.     If  cylinder  (c)  were  oblique,  an 
oblique  aux.  view  could  be  used. 

(d)  The  simplest  method  of  determining  the  intersection  in  Fig.   148  is  to 
pass  C.  Ps.  through  the  elements  of  (B).      The  sees,  of  (B)  will  thus  be  As  and 
those  of  (A)  ellipses,  or  other  curves. 

(e)  Fig.  149  represents  a  form  similar  to  the  stub  end  of  a  connecting  rod. 
The  highest  pts.,  B  and  D,  of  the  curves,  in  which  the  bell-shaped  neck  of  the 


70 


KSSKNTIAI.S    <>l     M  I .( 'I  I  \M<  A  I.    DKA1T1.V, 


cylindric  portion  intersects  the  rectangular,  are  directly  determined  in  the  front 
and  side  views,  and  will  be  seen  from  the  plan  to  be  the  pts.  in  which  circular 
sees,  become  tangent  to  the  sides  of  rectangular  sees.  The  lowest  pts.,  A,  C, 
E,  and  F,  lie  in  the  O  through  the  corners  of  the  rectangle  and,  therefore,  located 
in  the  front  and  side  views  by  projecting  from  the  plan  to  the  corresponding 
views  of  that  O ,  which  are  obtained  as  shown. 

Other  pts.  are  determined  by  intermediate  C.  Ps.  Thus  plane  G-H  gives 
a  O  which  cuts  the  rectangle  at  1,  2,  3,  and  4  of  the  required  curve,  which  are 
located  in  the  other  views  as  in  the  preceding. 


I 


FIG.  149 


CHAPTER  VIII 
DEVELOPMENT   OF   SURFACES 

81.  Principles  and  Methods.  A  development  is  the  representation  of  the 
surfaces  of  an  object  as  laid  out,  unfolded,  or  unrolled  into  the  plane  of  the  draw- 
ing. The  operation  is  suggested  pictorially  by  Figs.  150  (a),  (b).  A  develop- 
ment thus  shows  the  exact  area  of  all  surfaces  of  the  object  and  the  exact  length 
of  every  line  of  those  surfaces,  Fig.  151.  Plane,  cylindric,  and  conic  surfaces 
only  can  thus  be  developed.  Surfaces  of  double  curvature  and  warped  surfaces 
may  be  developed  approximately  by  assuming  portions  to  be  cylindric,  conic, 
or  triangular. 


FIG.  150 


Developments  are  made  to  determine  the  shapes  of  surface  patterns  required 
in  constructing  objects  of  sheet  metal,  cardboard,  etc.;  to  plot  groove  outlines 
for  cylindric  cams;  and  to  obtain  templets  or  patterns  for  irregular  surfaces,  etc. 

(a)  To   OBTAIN   A   DEVELOPMENT.      First    draw    the  views  from   which   the 
measurements  of  the  lines  can  be  made.     The  surface  may  be  imagined  as  opened 
on  any  lines  but  its  different  parts  should,  so  far  as  practicable,  be  represented 
as  attached  to  each  other, — each  being  so  placed  that  if  the  dev.  were  cut  out 
upon  its  outer  line  and  properly  bent,  it  would  form  or  envelope  the  object, 
surface  for  surface,  line  for  line. 

In  obtaining  lengths  from  the  views,  transfer  the  measurements  with  dividers. 

Remember  that  a  line  shows  its  true  length  only  upon  a  plane  to  which  it 
is  ||.  If  not  thus  shown,  its  true  length  must  first  be  determined.  The  dev.  may 
be  placed  in  any  convenient  position.  In  some  cases  it  is  possible  to  place  it 
so  that  one  set  of  dimensions  may  be  projected,  as  in  Fig.  151.  It  is  sometimes 
desirable  to  attach  it  to  some  line  or  lines  of  a  view  (Figs.  154,  159-161),  prac- 
tically as  revolved  about  those  lines. 

(b)  The  reproduction  of  forms  in  thick  paper,  by  cutting  and  bending  devs., 
will  prove  an  excellent  aid  to  correct  solutions.     The  surfaces  may  be  held  in 
position  by  providing  paste  laps  (see  Fig.   151)  or  by  using  gummed  binding. 
To  obtain  neat  edges  the  folding  lines  should  be  lightly  scored  with  a  sharp- 
pointed  knife. 


72  ESSENTIALS  OF   MECHANICAL   DRAFTING 

(c)  In  practical  work,  allowance  must  be  made  for  seams,  laps,  thickness 
of  material,  etc.  Economy  in  cutting  is  also  important. 

Material  for  double  curved,  or  warped  surfaces,  is  cut  to  patterns  of  assumed 
cylindric,  conic,  or  triangular  portions  and  then  beaten,  pressed,  or  stretched  to 
the  required  form.  In  some  cases  the  material  is  stamped  by  dies,  or  spun  to 
form  in  a  lathe. 

82.     Objects  Having  Plane  Surfaces. 

(a)  PRISMS.  The  dev.  of  a  right  p/ism  consists  of  two  similar  polygons  for 
the  bases  and  as  many  rectangles  as  the  prism  has  sides.  In  obtaining  the  dev. 
of  the  hexagonal  prism  (Fig.  151),  observe  that  the  lateral  edges  are  ||  to  V,  and 
the  base  edges  to  H.  The  front  and  rear  base  edges  are  also  ||  to  V. 

First  set  off  the  true  lengths  of  the  lower  base  edges  upon  a  line,  A-A.  At 
pts.  A,  B,  C,  etc.,  draw  J_s.  Upon  these  set  off  the  lengths  of  the  vert,  edges 
and  connect  the  pts.  to  complete  the  lateral  surface.  Attach  the  bases  to  any  side. 


7T-     I      6-E" 


^ 

J*- 

' 

6" 

v       B« 

F1* 

c 

Ev 

PASTE    LAPS 


FIG    151 


To  develop  the  prism  after  the  portion  above  an  oblique  C.  P.  1-4,  has 
been  removed,  obtain  base  line  A-A,  lower  base,  and  J_s  as  in  preceding.  Now 
assuming  the  surface  to  be  opened  from  edge  A-l,  this  edge  will  be  the  outer 
verts.  A-l,  in  the  development.  Beginning  with  A-l,  transfer  the  lengths  of 
the  verts,  to  the  corresponding  lines  in  the  dev.  Similarly  transfer  lengths 
of  base  edges  G-4  and  G-5  to  the  hor.  through  G.  The  lines  joining  1,  2,  3,  4, 
and  5,  6,  7,  1  will  be  the  dev.  of  the  line  of  intersection  of  the  C.  P.  with  the 
lateral  surface,  and  must,  therefore,  agree  in  length  with  the  corresponding  lines 
of  the  sectional  view.  Such  lines  should  always  be  compared ,  with  dividers. 
The  method  of  obtaining  the  portion  G435  of  the  upper  base  is  evident.  To 
complete  the  dev.,  draw  a  figure  similar  and  equal  to  the  true  shape  of  the  oblique 
surface  by  Art.  56,  attaching  it  to  any  side,  as  B  C  3  2. 


DEVELOPMENT   OF  SURFACES  73 

(b)  PYRAMIDS.  The  dev.  of  a  right  pyramid  consists  of  a  polygon  for  the 
base  and  as  many  As  as  the  pyramid  has  sides.  The  slant  edges  of  the  rect- 
angular pyramid  (Fig.  152)  are  equal,  but  are  not  projected  in  their  true  length 
in  either  view.  Assuming  the  surface  to  be  opened  from  one  of  these,  obtain  its 
true  length  as  in  Art.  71,  and  with  this  as  rad.,  describe  an  indefinite  arc  DEBCD. 
Upon  the  arc  set  off  the  successive  lengths  of  the  base  edges.  Join  these  pts. 
and  each  to  the  center  A,  the  vertex,  to  complete  the  lateral  surface.  Then 
attach  the  base.  Or,  lay  off  first  one  side  of  the  base  as  E-B,  and  with  E  and 
B  as  centers  and  the  true  length  of  the  slant  edges  as  rad.,  describe  arcs  to  inter- 
sect at  A;  then  proceed  as  in  preceding. 

Since  the  altitude  of  each  side  is  JL  to  a  base  edge,  the  vertex  A  could  be 
determined  by  setting  off  the  true  length  of  an  altitude,  as  A-F,  J_  to  its  corre- 
sponding bas§  edge  E-B,  as  indicated. 

In  pyramids  whose  inclined  edges  are  not  equal,  it  is  necessary  to  obtain 
the  true  length  of  each  separately, and  to  construct  the  As  by  Art.  37,  joining 
them  on  their  common  sides. 


To  develop  the  pyramid  (Fig.  152)  when  truncated,  first  obtain  dev.  of  entire 
pyramid.  Then  obtain  the  true  distances  of  pts.  1,  2,  3,  and  4,  from  the  vertex, 
or  base  (Art.  71).  Set  off  these  distances  upon  the  corresponding  lines  in  the 
dev.  and  join  the  pts.  thus  found  to  complete  the  sides.  Finally,  copy  the  true 
shape  of  the  top  surface  from  the  sectional  view,  attaching  it  to  any  side,  as 
BC21. 

83.     Objects  Having  Cylindric  or  Conic  Surfaces. 

(a)  CYLINDERS.  Since  the  elements  of  a  right  circular  cylinder  are  equal 
and  _|_  to  the  bases,  the  dev.  of  the  curved  surface  (Fig.  153)  will  be  a  rectangle 
whose  height  is  equal  to  the  length  of  the  elements,  and  base  equal  to  the  length 
of  the  circumference. 

The  length  of  the  base  edge  may  be  obtained  as   in    Art.  32,    and  laid  off 


71 


ESSENTIALS  OK  MECHANICAL  DRAFTING 
FIG   153 


,^rrh> 


upon  a  line  A-A.  At  the  ends  of  this  line  draw  _Ls  and  complete  the  rectangle. 
As  each  of  the  bases  would  touch  the  dev.  of  the  curved  surface  at  but  one 
pt.,  it  is  not  necessary  to  attach  them.  / 

To  develop  the  curved  surface  when  cut  off  on  line  12341,  obtain  base 
line  A-A,  divide  it  into  12  or  24  equal  parts  according  to  the  number  of  elements 
assumed  upon  the  cylinder,  and  through  the  pts.  draw  J_s.  The  length  of  the 
cut  elements  may  then  be  transferred  from  the  front  view  as  in  the  case  of  the 
edges  of  a  prism.  The  curve  traced  through  1,  2,  3,  4,  1  will  complete  the  lateral 
surface.  See  Art.  18(a). 

When,  as  in  (B)  Fig.  159,  neither  end  is  a  O,  both  will  develop  as  curves. 
It  is  necessary,  therefore,  to  assume  some  O  of  the  surface  whose  development 


Profile  of 
D  and  B' 


FIG.  1 54 


Pattern    for    F     Use   right  half   for  B. 
I     /left  half    for   B' 
x;    .  A/  DETAIL   OF 

\\-  -ty  PILASTER    CAP 

Make   4       16  Oz  Copper 


DEVELOPMENT   OF   SURFACES 


75 


may  be  used  as  a  base  line 
upon  which  to  set  off  the  true 
distances  between  the  ele- 
ments and  from  which  to  set 
off  their  lengths.  Such  line 
may  be  obtained  by  a  C.  P., 
a-Ev,  taken  JL  to  the  ele- 
ments as  shown.  See  also 
Fig.  160. 

Fig.  154  illustrates  the 
dev.  of  an  object  with  cy- 
lindric  curved  sides.  The 
true  distances  between  the 
elements  of  surfaces  (A) 
and  (B)  are  determined 
upon  A-E  and  L-O  in  the 
elevation  and  their  lengths  in 
the  plan.  The  distances  between  the  elements  of  (E)  and  (F)  are  identical  with 
those  of  (A)  and  (B).  To  determine  the  true  distances  between  the  elements 
of  (C)  and  (D)  it  is  necessary,  since  these  elements  are  not  _L  to  V,  H,  or  P, 
to  obtain  the  profiles  of  (C)  and  (D)  by  aux.  views  or  sees,  upon  planes  _L  to  the 
elements,  as  indicated. 

(b)  CONES.  Since  the  elements  of  a  right  circular  cone  are  of  equal  length, 
the  dev.  of  the  curved  surface  (Fig.  155)  will  be  a  sector  whose  radial  sides,  A-D, 
are  equal  to  the  true  length  of  an  element  and  whose  arc,  D-D,  equals  the  length 
of  the  circumference  of  the  base.  To  determine  the  length  of  this  arc,  obtain 
TT  or  T?  of  the  base  circumference  and  set  it  off  12  or  24  times  in  the  dev. 

To  develop  the  portion  below  the  parabolic  sec.,  obtain  the  base  arc  D-D, 
divide  it  into  12  or  24  equal  parts  according  to  the  number  of  elements  assumed 
upon  the  cone  and  draw  the  elements.  Then  find  their  lengths  when  cut  off, 
as  in  the  case  of  the  pyramid,  Art.  82(b). 

Since  the  left  element  is  ||  to  V,  the  true  distance  of  pt.  1  from  the  vertex 
A  is  seen  in  the  front  view.  The  true  distances  from  A  of  intermediate  pts., 
as  2  and  5,  are  not  seen  in  either  view,  but  as  the  elements  of  the  cone  are  equal, 
hors.  from  these  pts.  to  A-B  will  determine  upon  the  latter  the  required  distances. 
(See  Art.  71(b).)  Having  transferred  these  to  corresponding  elements  in  the 
dev.,  trace  the  curve  through  them. 

The  method  of  developing  a  right  circular  conic  surface  whose  ends  are  not 
in  planes  _L  to  its  axis  is  evident  from  Fig.  161  (b). 

In  a  conic  surface  other  than  right  circular  as  in  Fig.  156,  or  one  whose  vertex 
is  inaccessible  as  in  Fig.  161,  it  is  necessary  to  assume  the  surface  to  be  composed 
of  plane  As  whose  sides  are  elements  and  whose  bases  are  short  chords  of  the  base 
of  the  cone.  The  method,  called  development  by  triangulation,  is  thus  identical 
with  that  used  in  developing  a  pyramid  with  unequal  slant  edges.  Art.  82 (b). 

To  develop  the  oblique  elliptic  cone  (Fig.  156)  obtain  equidistant  elements 
as  shown.  Assuming  the  surface  to  be  opened  upon  A-D,  the  line  A-B  will  be  the 


7.1 


ESSENTIALS  OF   MECHANICAL   DRAFTING 


C.  L.  of  the  dev.  and  may  be  drawn  directly  at  A'-B',  equal  to  its  true  length 
AV-BV.  The  true  lengths  of  elements  A-l,  A-2,  etc.,  may  be  found  by  revolving 
them  ||  to  V.  To  avoid  confusing  the  views,  however,  a  separate  diagram  of  true 
lengths  may  be  constructed.  The  true  length  of  any  element  will  be  the  hypot- 
enuse of  a  right  A  whose  base  is  equal  to  the  length  of  that  element  in  the  plan 
and  its  altitude  equal  to  the  vert,  height  of  one  end  pt.  above  the  other  in  the  ele- 


3F\  \     \ xN^True  Lengths 

/'    /       '  '  V* t      \\  \^    \\ 

^  /  /    !   \\      _aX\\V--  r'^156 

T        F  C?          3VIX   4V  Ov        IX       D'  '4-  13     Ic     12(18 


FIG.  157 


vation.  Hence,  draw  A-X  _L  to  the  base  line  and  equal  to  the  altitude  of  the  cone; 
then  lay  off  X-l,  X-2,  etc.,  equal  to  AH-1H,  AH-2H,  etc.,  and  connect  pts.  1,  2, 
etc.,  to  A.  Observe  that  this  is  equivalent  to  revolving  the  elements  ||  to  P.  Now 
with  A'  as  center  and  rad.  A-l  draw  an  arc  across  A'-B';  with  B'  as  center  and 
rad.  BH-1H  intersect  this  arc  at  1'.  Then  A'-l'  will  be  the  developed  position 
of  A-l.  Continue  this  operation  until  all  elements  have  been  determined  and 
trace  the  curve  B'1'2',  etc.,  through  the  base  pts.  as  indicated. 

The  figure  also  illustrates  the  method  of  developing  the  cone  when  truncated. 


DEVELOPMENT  OF  SURFACES 


77 


84.     Objects  Having  Double  Curved,  or  Warped  Surfaces. 

(a)  Fig.  157  illustrates  the  method  of  developing  a  double  curved  surface 
by  assuming  portions,  as  A  7  8,  to  be  cylindric,  also  by  assuming  portions  between 
the  O  s  to  be  conic. 

Fig.  158  illustrates  a  transition  piece  of  piping,  A,  whose  surface  is  warped, 
being  neither  cylindric  nor  conic.  Such  surfaces  can  be  developed  approxi- 
mately only,  by  triangulation.  Art.  83 (b). 


(b)      I 


FIG  158 


Half    Pattern     for  A 

measurable  As  as  shown.  In  the  plan 
divide  half  of  the  base  and  curved  end 
of  the  top  edge  into  the  same  number 
of  equal  parts,  at  least  six.  Observe 
that  as  the  planes  of  both  edges  are  ob- 
lique to  H,  each  must  first  be  revolved 
|j  to  H,  or  V,  and  then  back  to  its 
oblique  position.  Join  the  corre- 
sponding pts.  in  each  with  full  lines 
1-A,  2-B,  etc.;  also  draw  lines  2-A,  3-B, 
etc.,  dotted  for  contrast. 

Next  construct  the  diagram  (a)  for 
the  true  lengths  of  the  full  lines,  noting 

that  neither  the  upper  nor  lower  pts.  are  on  the  same  levels.     Similarly  construct 

diagram  (b)  of  true  lengths  for  the  dotted  lines. 

As  the  surface  is  symmetrical  about  the  C.  L.  1-Y,  the  line  1-X  may  be  drawn 

directly  at  l'-X'  in  the  dev.  (c),  equal  to  its  true  length   1V-XV.      With   1'  as 


78  ESSENTIALS  OF  MECHANICAL  DRAFTING 

center  and  rad.  1-A  of  diagram  (a)  describe  an  arc  across  l'-X'.  With  X'  as 
center  and  rad.  X-A  of  the  plan,  intersect  this  arc  at  A'.  Then  l'-A'  will  be  the 
developed  position  of  line  1-A.  With  A'  as  center  and  rad.  A-2  of  diagram  (b), 
describe  an  arc.  With  1'  as  center  and  rad.  1-2  of  the  revolved  base,  intersect 
this  arc  at  2'.  Then  A'-2'  will  be  the  developed  position  of  line  A-2. 

With  2'  as  center  and  rad.  2-B  of  diagram  (a),  describe  an  arc.  With  A' 
as  center  and  rad.  A-B  of  the  revolved  top  edge,  intersect  this  arc  at  B'.  Then 
2'-B'  will  be  the  developed  position  of  2-B.  Continue  these  operations  until 
all  points  have  been  determined  and  complete  the  dev.  as  shown. 

85.  Intersecting  Surfaces.  In  intersecting  parts  of  an  object,  the  line  of 
intersection  being  common  to  both  surfaces  will  be  developed  in  each.  As 
this  line  is  determined  by  the  pts.  in  which  the  edges,  elements,  or  other  lines 


FIG.  159 


of  each  part  intersect  the  surfaces  of  the  other,  and  the  dev.  gives  the  true  length 
of  every  line  of  every  surface,  it  follows  that  these  pts.  may  be  found  by  trans- 
ferring the  true  lengths  of  the  lines  which  determine  them  in  the  views,  to  the 
corresponding  lines  drawn  in  the  dev. 

(a)  The  lateral  surface  of  (B)   (Fig.   159)  is  developed  as  explained  in  Art. 
83 (a).     The  dev.  of  the  rear  half  only  is  shown  and  is  attached  upon  element 
A-B  for  convenience  in  transferring  lengths.     The  distances  between  the  elements 
are  seen  in  the  revolved  sec.  of  a-Ev  and  transferred  to  the  base  line  E-a,  as  shown. 

(b)  The  method  of  developing  the  lateral  surface  of  (C)  is  evident.     Lines 
as  1-2  and  3-4,  which  give  intermediate  pts.  of  the  line  of  intersection,  are  found 
by  obtaining  their  true  distances  from  the  ||  edges  in  the  revolved   sec.,    and 
their  true  lengths,  from  either  the  front  or  top  view.     The  dev.  of  the  upper 
half  only  is  shown. 


DEVELOPMENT  OF  SURFACES 


79 


(c)  To  develop  the  lateral  surface  of  (A),  first  obtain  the  dev.  of  the  entire 
surface,  that  is,  a  rectangle  (rear  half  only  is  shown).  Assume  surface  to  be 
opened  from  right  element.  This  element  cuts  the  surface  of  (C)  in  I  and  M. 
The  opposite  element  cuts  the  surface  of  (B)  in  F  and  B  and  will  be  in  the  center 
of  the  dev.  From  these  elements  the  positions  of  the  other  elements  of  (A) 


which  cut  the  surfaces  of  (B)  and  (C)  may  be  determined.  The  distances  of 
these  elements  from  the  outer  elements  are  seen  in  the  top  view,  and  the  distances 
of  their  points  of  intersection  from  the  bases  of  (A)  are  seen  in  the  front  view, 
(d)  Fig.  160  illustrates  a  pipe  elbow  with  conic  and  pyramidal  flanges;  Fig. 
161,  two  other  common  forms,  the  methods  of  developing  which  are  indicated. 


FIG.  161 


CHAPTER  IX 
MECHANICAL    PICTORIAL    DRAWINGS 

86.  Character  and  Purpose  of  the  Drawing.  It  has  been  noted  that  two  views 
at  least  are  required  to  show  the  exact  form,  size,  and  relation  of  all  lines  and 
surfaces  of  an  object,  and  that  two  only  of  its  three  dimensions  can  be  shown 
in  their  actual  proportion  and  relation  in  one  view.  It  is  sometimes  necessary, 
however,  to  represent  an  object  or  detail  by  a  single  oblique  view  having  a  more 
or  less  pictorial  effect,  while  at  the  same  time  showing  the  relative  proportions 
of  its  principal  lines  or  dimensions  to  a  scale.  Compare  Figs.  162  (a),  (b),  (c), 
etc.;  163  (a)  and  (b);  169  (a)  and  (b). 


FIG.  162 


(c) 


(d) 


\ 


F  D 

ORTHOGRAPHIC 
PROJECTION 


ISOMETRIC 
PROJECTION 


D 

OBLIQUE    PROJECTIONS 


0  D 

TRUE.    PICTORIAL     REPRESENTATIONS 

Such  representations  are  made  in  place  of  or  supplementary  to  the  usual 
views  for  the  purpose  of  general  illustration,  or  to  describe  details  of  machine, 
furniture,  and  building  construction  and  assembly  more  directly  and  clearly. 
They  are  also  extensively  employed  for  catalog  illustrations  and  Patent  Office 
drawings. 

The  methods  ordinarily  used  are  those  of  Isometric  Projection  and  Oblique 
Projection,  the  choice  depending  upon  the  form  of  the  object  and  the  particular 
effect  desired  to  be  given.  A  drawing  made  by  either  method  is  never  a  true 
picture,  since  ||  lines  are  always  represented  by  ||s  as  in  orthographic  proj., 


MECHANICAL  PICTORIAL  DRAWINGS  81 

while  in  a  true  pictorial  representation  the  apparent  convergence  of  receding 
||s  is  shown.  Hence,  such  drawings  often  give  an  unsatisfactory  effect  of 
•distortion. 

87.  Isometric  Projection.  If  a  cube,  placed  as  shown  in  Fig.  162(a),  be 
turned  45°  about  a  vert,  axis,  then  forward  until  the  diagonal  of  the  cube 
through  A  is  ±  to  V,  all  of  the  edges  will  be  equally  inclined  to  that  plane,  and 
being  equal  in  length  their  projs.  upon  it  will  be  equal.  Such  view  is  an  isometric 
(equal  measure)  projection  of  the  cube.  (See  Fig.  (b).) 

When  a  rectangular  object  is  thus  related  to  the  plane  of  proj.,  it  will  be 
seen  that  each  of  its  edges  will  be  ||  to  one  or  another  of  three  mutually  JL  lines, 
as  A-B,  A-C,  and  A-D,  which  correspond  in  direction  to  the  axes  or  principal 
dimensions  (1,  b,  and  t)  of  the  object.  These  lines  are  called  the  isometric  axes, 
and  their  projs.  form  equal  Zs  of  120°  about  a  pt.  A.  All  lines  of  the  object 
coinciding  with  or  1 1  to  the  isometric  axes  are  called  isometric  lines;  all  others  are 
non-isometric  lines.  The  planes  determined  by  the  isometric  axes  and  all  planes 
||  to  these  are  called  isometric  planes. 

Note  that  isometric  lines  only,  make  equal  Zs  with  the  plane  of  proj.;  hence, 
equal  distances  upon  such  lines  only  will  be  equally  foreshortened  in  the  view. 

(a)  Instead  of  obtaining  the  view  by  the  methods  of  Art.  72,  an  isometric 
is  usually  constructed  by  setting  off  the  limiting  pts.  of  the  lines  directly  upon 
isom.  lines  and  joining  the  points  thus  determined.  Measurements  must  never 
be  made  upon  non-isometric  lines.  The  foreshortening  of  the  isometric  lines, 
which  is  about  .81  of  full  size,  is  usually  disregarded  and  the  measurements 
made  equal  to  the  true  lengths  of  the  corresponding  lines  of  the  object,  or  to 
some  other  common  scale. 

One  of  the  axes  is  usually  assumed  as  vert. ;  the  others,  therefore,  are  at  30° 
with  the  hor.  direction,  as  in  Fig.  162 (b).  When  the  lower  surface  of  an  object 
is  to  be  visible  the  axes  are  reversed.  Invisible  lines  are  omitted  unless  they 
give  necessary  information. 

When  dimensions  are  given,  the  dimension  and  extension  lines  must  be 
isometric.  See  Fig.  163. 

In  general,  sections  and  breaks  should  be  taken  in  isom.  planes.  See  Figs. 
164,  170,  194;  also  invisible  section,  Fig.  165. 

88.     To  draw  the  isometric  of  a  rectangular  object. 

(a)  A  CUBE.     Fig.  162(b).     From  any  pt.,  as  A,  draw  indefinite  lines  A-B 
and  A-C  at  30°  and  a  vert.  A-D.     Upon  these  set  off  the  given  dimensions  of 
the  object  to  the  desired  scale.      Since  each  edge  is  ||  to  one  or  another  of  the 
isom.  axes,  it  will  be  ||  to  the  corresponding  line  in  the  drawing,  and  the  rep- 
resentation may  be  completed  as  shown.     It  is  evident  that  the  isom.  could  be 
started  from  any  assumed  pt.,  as  D  or  E. 

(b)  AN  OBJECT  WITH  RECTANGULAR   DETAILS.     Fig.   163.     Starting  at  any 
pt.,  as  A,  draw  the  isom.  of  the  bottom  board  (A),  obtaining  all  measurements 
from  the  front  and  top  views. 


s-J 


ESSENTIAL   ol     MICHANICAL   DRAFTING 


j 

v 

-i 

^ 

' 

r 

-= 

1 

: 

H        f 

r 

J_ 

Tff 

—  f-I— 

1 

F 

G     JH   -^ 

(a) 

i. 

7^ 

C 

1  * 

7« 

. 

is 

4 

G4- 

1     { 

n 

-i 

-1  T  5 

1 

1 
CM  „ 

\      h 

• 

G     ± 

c.l 

(b) 


1 

I 

r 

FIG.  164 


FIG.  165 


MECHANICAL   PICTORIAL   DRAWINGS 


S3 


Details  must  be  built  up  from  the  surface  or  surfaces  which  they  intersect. 
Thus  to  draw  block  (B),  locate  a  corner,  as  E,  and  draw  the  line  of  intersection 
E  1  2  3;  then  proceed  practically  as  with  (A). 

To  draw  (C),  one  of  its  lower  corners,  as  F,  must  first  be  located  upon  the 
top  face  of  (A)  by  means  of  co-ordinates  B-4  and  4-F;  that  is,  by  measuring  the 
distance  of  F  from  B  along 

lines   which  will   be   ||   to          \  XI      (a) 

two  of  the  isom.  axes.  To 
draw  the  recess  in  the  front 
of  (C),  locate  a  corner, 
as  G,  by  co-ordinates  F-5 
and  5-G,  and  proceed  as 
before.  Corner  H  is  de- 
termined in  like  manner. 

89.  To  draw  the  iso- 
metric of  an  object  involv- 
ing non-isometric  figures. 


(b) 


A.I        FIG,  166 


(b) 


The  form  and  position  of  an  integral  part  of  an  object  are  frequently  such  that 
some  or  all  of  its  lines  will  be  non-isom.  Such  lines  are  determined  in  the  isom. 
by  co-ordinates  1 1  to  two  or  to  all  three  of  the  axes. 

(a)  STRAIGHT  LINES.     In  Fig.  165    each  of  the  inclined  lines  in  (a)  will  be 
oblique  to  two  of  the  axes  and  hence  determined  by  co-ordinates    ||    to  those 
axes,  as  shown.     See  also  oblique  lines  in  Figs.  171,  172,  194. 

In    Fig.    166   the    line 

A-B  will  be  oblique  to  __  ^  \^  ^  (°) 
the  three  axes.  The  end 
B,  was  located  in  this  case 
by  co-ordinates  A-l,  1-2, 
2-B  ||  to  those  axes.  The 
end  pts.  of  C-D  were  de- 
termined in  like  manner. 

(b)  POLYGONS.    In  Fig. 
167  two  sides  only  of  the 
hexagon  ABCDEF  will  be 
isom.      Each    pt.  may  be 
referred    by    co-ordinates 
to  two  of  the  isom.  axes, 
as  in  (a),  or  to  the  sides 
to  those  axes. 

By  placing  this  rectangle  with  one  of  its  lines    ||    to  or  coincident  with  its 
isom.,  one  set  of  dimensions  may  be  projected  to  the  required  figure,  as  shown. 

(c)  CURVES.      The   method   of  determining  a  curve   in  an  isom.    drawing  is 
identical  with  that  of  a  rectilinear  figure  save  that,  in  the  absence  of  vertices, 
pts.  must  be  assumed  in  the  curve  and   referred  to  the  axes,  as  illustrated  in 
Figs.  168(a),   169,  170. 


FIG.  167  2V 

of  a   circumscribing   rectangle   whose   sides  will  be 


-1 


ESSENTIALS  OF  MECHANICAL  DRAI-TIN<; 


The  projection  of  a  O  in  an  isom.  plane  is  an  ellipse,  in  determining  which 
it  is  convenient  to  circumscribe  a  square.  Fig.  168(a).  As  the  axes  of  the  ellipse 
coincide  with  the  diagonals  of  the  isom.  square,  their  end  pts.  8,  6,  5,  and  7  could 
be  determined  by  co-ordinates  and  the  curve  described  by  Art.  64(b).  See 
also  Art.  64 (d),  Note  2. 


C     D 


; 


(b) 


FIG   168 


An  approximate  method  of  drawing  the  ellipse  by  circular  arcs,  which  is 
usually  sufficiently  exact,  is  shown  in  Fig.  168(b).  The  centers  are  the  inter- 
sections of  J_s  to  the  sides  of  the  square  at  the  middle  pts. 

The  application  of  this  method  to  the  rounding  of  corners  is  shown  in  Fig. 
169.  The  construction  at  D  determines  the  radii  for  all. 

To  draw  a  curve  not  in  an  isom  plane,  as  A-B  (Fig.  170),  determining  pts. 
must  be  located  by  co-ordinates  ||  to  the  three  axes,  as  indicated.  Screw 
threads  are  usually  represented  conventionally  as  in  Fig.  173. 


(b) 


FIG.  169 


(d)  SOLIDS.  The  method  of  drawing  an  object  by  inscribing  it  in  an  isom. 
solid  is  evident  from  Figs.  167,  171.  When  it  cannot  be  thus  inscribed,  pts. 
must  be  referred  to  the  axes  as  indicated  in  Fig.  166. 

In  order  to  preserve  the  symmetrical  appearance  of  the  piece  in  Fig.  172, 
it  was  necessary,  either  to  draw  the  top  view  turned  through  45°,  or  to  take  the 
hor.  measurements  for  the  groove  centers  on  45°  lines  instead  of  on  hors.  and  verts., 


MECHANICAL   PICTORIAL   DRAWINGS 


85 


(b) 


r 


FIG  171 


(b) 


I— 1  Y 


FIG. 172 


St. 


ESSENTIALS  OF   MECHANICAL   DRAFTING 


as  shown.     Observe  that  the  outer  elements  of  conic  surfaces  would  be  tangent 
to  the  ellipses  and  not  drawn  to  ends  of  the  axes.     See  A  and  B,  Fig.  173. 

To  determine  the  outlines  of  surfaces  of  double  curvature,  obtain  a  series 
of  sections  and  draw  the  required  curve  tangent  to  the  isom.  of  these,  as  indicated 
in  Figs.  174,  175,  also  149. 

90.  Oblique  Projection.  An  oblique  projection  is  obtained  by  means  of 
||  projectors  oblique  to  the  plane,  the  object  being  so  placed  that  two  of  its 
axes  or  principal  dimensions  are  ||  to  the  plane.  In  the  case  of  a  rectangular 
object,  as  a  cube,  the  view,  therefore,  gives  the  true  shape  and  size  of  its  front 
and  rear  faces  and  two  dimensions  of  the  object.  All  lines  _l_  to  the  plane  are 
projected  as  ||  lines  whose  direction  and  lengths  depend  upon  the  direction  of 
the  projectors.  They  are  usually  drawn  at  30°,  45°,  or  60°,  up  or  down  to  left 


FIG  173 


or  right,  and  made  equal  to  the  full  scale  length  of  the  corresponding  lines  of 
the  object  as  in  Fig.  176,  or  foreshortened,  usually  half,  as  in  Fig.  177.  The 
latter  gives  a  better  pictorial  effect,  but  requires  the  use  of  two  scales,  as  is  evident. 

When  the  receding  _Ls  are  foreshortened  one-half  and  inclined  at  45°,  the 
view  is  sometimes  called  a  cabinet  projection. 

The  axes  in  oblique  proj.  are  thus  a  vert.  A-B,  a  hor.  A-C,  and  a  line  A-D 
at  30°,  45°,  or  other  Z. .  Measurements  must  be  made  only  upon  these  axes 
or  lines  ||  to  them.  Curves,  and  lines  not  ||  to  the  axes,  must  be  determined 
by  co-ordinates  ||  to  the  axes,  as  in  isom.  drawing. 

A  circle  not  ||  to  the  plane  of  proj.  will  be  projected  as  an  ellipse,  which  may 
be  obtained  as  in  Art.  89(c). 


MECHANICAL   PICTORIAL   DRAWINGS 


87 


An  obvious  advantage  of  oblique  proj.  over  isom.  lies  in  the  possibility  of 
representing  certain  curved  and  irregular  surfaces  in  their  exact  shape  and  size. 
See  applications  of  principles  in  Figs.  178,  181. 


(b) 


FIG  175 


FIG.  176 


FIG. 177 


(a) 


L> 


FIG.  178 


91.  Shade  Lines,  Shadow  Lines,  and  Line  Shading.  In  finishing,  visible 
edges  between  light  and  dark  surfaces  are  frequently  shaded,  as  in  Figs.  162(b), 
164,  167,  169,  178(a).  The  indication  of  these  shade  lines  adds  relief  to  the 
drawing  and  increases  its  pictorial  effect. 


ss  ESSENTIALS  OF   MECHANICAL   DRAFTING 

In  isometric  drawing  the  rays  of  light  are  assumed  ||  to  the  plane  of  proj.  and 
at  30°  down  to  the  right.  Hence,  all  rectangular  objects  and  parts,  whose  lines 
are  ||  to  the  axes,  have  their  shade  lines  in  the  same  relative  positions  as  in  the 
cube.  Fig.  162(b). 

In  oblique  proj.  the  rays  are  assumed  ||  to  the  diagonal,  C-E,  of  the  cube. 
Fig.  162(c). 

In  determining  the  shade  lines,  the  shadows  are  disregarded.  Elements  of 
curved  surfaces  are  generally  not  shaded.  Edges  of  cylindric  surfaces  may  be 
shaded  as  in  Figs.  169,  178(a). 

Instead  of  shade  lines,  shadow  lines  may  be  applied  as  in  Art.  23.  (See 
Figs.  171,  178(b).)  Another  method  is  to  shade  the  nearest  edges  as  in  Fig.  172. 
Line  shading  may  be  applied  as  in  Art.  24. 


CHAPTER  X 
WORKING   DRAWINGS 

92.  Character  and  Purpose  of  the  Drawing.     A  working  drawing  is  a  mechan- 
ical drawing,  containing  all  information  as  to  form,  dimensions,  construction, 
material,    finish,    etc.,    necessary   to   the   workman   or   mechanic  in   making  or 
building  the  object  which  it  represents.     See  Art.  1. 

To  convey  this  information  readily,  the  drawing  must  be  accurate;  as 
clear,  simple,  and  direct  as  possible;  and  in  accordance  with  shop  and  drafting 
practice.  It  must  express  the  idea  completely  and  definitely  and  contain  noth- 
ing that  is  unnecessary,  ambiguous,  or  misleading.  In  commercial  drafting 
utility  of  the  drawing  and  economy  of  production  are  the  ends  sought  for  in 
all  cases. 

The  nature  of  the  views  of  which  the  drawing  is  composed  is  explained  in 
Chap.  V. 

93.  Types   of   Drawings.     In    complicated    objects    composed    of    different 
pieces  or  parts,  certain  features  would  inevitably  be  hidden  or  not  clearly  shown; 
hence,  in  such  cases,  two  types  of  drawings  are    required — namely,  general  or 
assembly  drawings  and  detail  drawings. 

(a)  The  purpose  of  a  general  drawing  (Figs.  179,  184,  185,  202)  is  to  illustrate 
the  design  of  the  subject  as  a  whole  and  to  show  the  relative  positions  of  the 
different  pieces  composing  it.     It  may  include  the  complete  description  of  some 
or  of  all  of  the  pieces,  or  give  merely  such  information  as  may  be  necessary  in 
assembling  them  or  erecting  the  object.     As  a  rule,  it  should  be  as  free  from 
representation  of  minor  detail  and  hidden  parts  as  possible. 

(b)  A  detail  drawing  (Figs.  180,  203-206,210-213)  shows  each  piece  by  itself 
and   gives   all   information    necessary   for   making   it.     In   simple   objects,    full 
instructions  would  ordinarily  be  given  in  a  general  drawing,  sometimes  called 
a  detailed  assembly.     See  Figs.  179,  184,  185. 

Note. — The  table  shown  in  Fig  179  was  separately  detailed  (Fig.  180)  for  purpose  of  comparison 
with  the  general  drawing  and  to  illustrate  certain  methods  of  arrangement,  dimensioning,  etc. 

Detail  drawings  may  be  shown  on  the  same  sheet  with  the  assembly,  grouped 
on  separate  sheets,  or  each  piece  shown  on  a  sheet  by  itself,  depending  upon 
the  character  of  the  object,  size  and  number  of  parts,  etc.  In  general,  details 
of  the  pieces  of  one  part  of  an  object  should  be  grouped  apart  from  similar  groups 
of  other  parts  (Fig.  205) ;  and  so  far  as  possible  the  arrangement  of  details  of 
related  pieces  should  be  such  that  reference  may  readily  be  made  from  one  to 
the  other.  Figs.  180,  205.  It  is  often  desirable  to  show  related  pieces  of  a 
part  as  assembled.  Figs.  181,  182. 

In  drawings  of  machinery  the  special  information  required  by  the  different 
workmen,  as  the  pattern  maker,  the  blacksmith,  and  the  machinist,  is  often 


ESSENTIALS  OF   MECHANICAL   DRAFTING 


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WORKING   DRAWINGS 


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ESSENTIALS  OF  MECHANICAL  DRAFTING 


detailed  upon  separate  sheets  for  each.     Likewise  work  required  to  be  done  on 
certain  machines,  etc.,  is  often  so  grouped. 

94.     Position  of  Object  and  Arrangement  of  Views. 

(a)  The  object  should  be  represented  in  its  natural  position,  and,  when  it 
has  a  definite  front,  that  part  should  be  shown  in  the  front  or  principal  view. 
The  position  of  a  separate  piece,  however,  should  be  such  as  to  show  best  its 
form,  with  the  fewest  lines  and  views,  regardless  of  its  position  in  the  complete 
object. 

(b)  In  general,  a  top,  side,  bottom,  or  rear  view  should  be  related  to  the 
front  view,  as  indicated  in  Fig.  120.     When  for  convenience  or  necessity  they 
are  not  thus  arranged,  it  is  well  to  letter  them:  "Top  View,"  "Side  View,"  etc. 
See  Figs.  181,  192.     Other   views   may   be   placed  where   most  convenient,  the 
part  to  which  each  refers  being  clearly  indicated  by  its  position  or  by  marking. 
See  Figs.  179,  182. 


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DETAILS    OF 

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FIG.  181 


95.  Selection  and  Number  of  Views,  etc.  Those  views  should  be  selected 
for  drawing  which  most  clearly  and  adequately  describe  the  object  and  require 
the  least  time  to  execute.  Views  that  do  not  add  clearness  or  convey  necessary 
information  should  be  omitted. 

Two  or  three  views  are  ordinarily  required;  others,  however,  are  frequently 
necessary.  In  simple  symmetrical  objects  one  is  often  sufficient.  Fig.  174(a). 

In  addition  to  the  external  views  the  following  are  frequently  necessary: — 

Sectional  views — to  show  interior  construction. 

Diagrams — to  show  the  direction  of  motion  of  moving  parts,  relations  of 
important  centers,  etc. 

Developments — to  show  true  shapes  for  surface  patterns,  templets,  etc. 

Isometric,  or  oblique  views — to  show  details  of  construction,  etc.,  with  a 
pictorial  effect. 


WORKING  DRAWINGS 


FIG.  182 


PILLOW    BLOCK 
AT  45° 


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|-z'«-{-«—  |-"Drill    13  Holes  at^"=  9" 


(a) 


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FIG.  183 


(b)  -a-  w/  - 

^"Drill.  12  Holes  |" Drill.  12  Holes 

Equally  Spaced 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


96.  Center  Lines.  As  a  rule,  symmetrical  objects  and  parts  should  have 
their  axes  and  centers  indicated  by  means  of  center  lines.  See  Art.  66(g). 
As  the  object  is  usually  represented  with  its  main  axes  ||  to  the  planes  of  the 
views,  the  main  or  principal  C.  Ls.  of  a  drawing  are  usually  vert,  and  hor.  lines 
passing  through  the  centers  of  the  corresponding  views  of  the  main  body  of 
the  object.  These  C.  Ls.  are  commonly  extended  to  connect  the  views. 

Secondary  C.  Ls.  are  likewise  usually  _|_  to  each  other  or  to  main  C.  Ls., 
but  are  ordinarily  not  extended.  When  the  centers  of  a  series  of  symmetrical 
parts  are  equidistant  from  a  common  center,  a  circular  C.  L.  is  used.  The  other 
C.  Ls.  for  these  parts  are  usually  radial  lines  from  the  center  of  the  circular  C.  L. 
Figs.  182-184,  188,  189. 

When  a  C.  L.  coincides  with  a  line  of  the  object,  the  latter  should  be  shown. 
See  Figs.  184,  220. 

A  straight  C.  L.  may  be  regarded  as  the  edge  view  of  a  center  plane. 


FLANGE  CO 


FIG.  184 


97.  Conventional  Representations.  Instead  of  true  or  complete  projections 
it  is  often  desirable,  for  clearness  or  economy  of  labor,  to  make  conventional  or 
approximate  representations.  Some  of  the  more  common  methods  used  are 
referred  to  in  the  following: — 

Partial  views,  as  of  one-half  or  other  suggestive  portion  of  an  object,  may 
often  be  used  in  place  of  complete  views.  Figs.  180,  182(a),  184,  188(a),  192-195, 
203-206. 

Lines  or  details  clearly  shown  in  one  or  two  views  may  often  be  omitted 
in  others,  especially  those  of  hidden  parts.  Figs.  184,  192,  193,  199. 

Of  a  series  of  similar  parts,  as  holes,  bolts,  etc.,  of  the  same  size,  it  is  usually 
necessary  to  draw  but  one  or  two  and  to  indicate  the  locations  of  the  others. 
A  brief  note  often  saves  the  drawing  of  many  lines.  Fig.  183. 

Screw  and  pipe  threads,  bolts  and  springs,  are  usually  represented  conven- 
tionally, as  in  Chapter  XI. 


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FIG.  185 


ESSENTIALS  OF   MECHANICAL   DRAFTING 


Ellipses  and  other  non-circular  curves  may  often  be  approximated  by  arcs 
of  0s.  Art.  18(b). 

Where  an  edge  is  rounded  so  that  no  definite  line  of  intersection  is  seen,  it 
is  often  better  to  show  such  line  as  if  existing.  See  curve  a-b,  Figs.  188,  208. 

In  Fig.  184,  instead  of  projecting  the  upper  bolt  from  the  circular  view,  it 
is  represented  at  its  true  distance  from  the  center  of  the  shaft,  the  same  as  the 
lower  bolt;  thus  avoiding  confusion  of  the  view  and 
suggesting  the  symmetry  of  the  piece.  For  similar 
reasons  the  lower  arm  of  the  wheel  (Fig.  188)  would 
be  shown  as  in  (b),  the  same  as  the  vert,  arm.,  instead 
of  foreshortened  as  in  (c).  See  also  grooves  in  Fig. 
172(a),  and  slots  and  ribs  in  Fig.  189. 

Other  conventions  are  described  in  Arts.  98  and  99.        p)G  135 

98.     Sectional  Views.     See  Chap.  VI. 

(a)  When  a  section  does  not  lie  in  a  main  center  plane,  the  place  where  the 
sec.  is  taken  should  be  indicated,  as  in  Figs.  179,  188(a),  192,  199.  Parts  lying 
beyond  the  sec.  need  not  be  shown  unless  they  add  clearness  or  give  addi- 
tional information.  When  lines  of  a  removed  portion  are  shown  they  may 
be  indicated  as  construction  lines.  See  top  view,  Figs.  179,  193,  199. 

Section  lines  should  have  the  same  direction  and  spacing  throughout  all 
parts  of  a  sec.  of  any  one  piece.  Fig.  188(b).  Different  pieces  in  a  sec.  are 
indicated  by  sec.  lining  in  opposite  directions,  at  different  Zs,  or  by  difference 
in  spacing.  Figs.  184,  185,  192.  Sec.  lines  must  never  cross  figures,  arrowheads, 
or  notes  placed  in  a  sec.  See  Fig.  184. 


BABBITT.  ETC  RUBBER.  ETC 


BRICK 


STONE 


CONCRETE 


FIG  187 


When  a  sec.  is  very  narrow,  it  is  sometimes  filled-in  black,  and  adjacent  pieces 
separated  by  narrow  spaces.  Fig.  186 (a).  Very  short  sec.  lines  may  be  drawn 
freehand.  In  large  drawings,  sees,  are  often  indicated  as  in  Fig.  186(b). 

(b)  MATERIAL  CONVENTIONS.  Different  kinds  of  materials  may  be  indicated 
by  different  kinds  of  sec.  lining.  The  conventions  shown  in  Fig.  187  are  com- 
monly employed,  but  there  is  no  fixed  standard  of  practice.  Many  draftsmen 
use  plain  sec.  lining  for  all  materials  and  indicate  the  kind  by  lettering.  On 
drawings  finished  on  paper  the  sees,  are  sometimes  tinted  with  India  ink  or  colors. 


WORKING   DRAWINGS 


97 


(c)  SELECTION  OP  SECTIONAL  VIEWS.  A  view  may  show  a  complete  section, 
that  is,  through  the  entire  object;  or  a  partial  section.  In  objects  symmetrical 
about  an  axis,  half  only  on  either  side  of  the  C.  L.  need  be  sectioned.  Figs. 
189(b),  192.  When  a  partial  sec.  is  not  limited  by  a  C.  L.  or  some  line  of  the 
object  it  is  usually  shown  as  in  Figs.  199,  218 (o). 

A  view  need  not  show  all  parts  in  sec.  that  lie  within  the  sec.  plane,  unless 
the  drawing  is  rendered  clearer,  or  additional  information  given.  In  general  a 
solid  cylindric  part  as  a  shaft,  rod,  bolt,  screw,  etc.,  intersected  by  a  plane  || 
to  its  axis,  should  be  shown  in  full;  likewise  a  key,  rib,  tooth,  wheel  arm,  or 
turned  handle. 


FIG.  188 


CONVENTIONAL        TRUE 
SECTION  SECTION 


In  Fig.  188  (a)  the  plane  A-A  passes  through  the  rim,  vert,  arm,  hub,  and 
handle  of  a  wheel.  Instead  of  making  a  true  sec.  (c),  as  projected  from  the 
view  (a),  the  draftsman  would  make  the  conventional  sec.  (b),  in  which  the  vert, 
arm  and  handle  are  shown  in  full.  He  would  also  represent  the  lower  arm  as 
in  the  same  center  plane  as  the  vert.  arm.  If  considered  by  itself,  the  true  sec. 
would  suggest  a  solid  web  between  the  hub  and  rim,  which  would  be  misleading, 
while  the  conventional  sec.  shows  the  desired  information  at  a  glance. 

The  conventional  sec.  of  a  ribbed  piece  is  shown  in  Fig.  189(b).  The  method 
(c)  is  also  used.  Note  location  of  slots  in  (b)  and  (c). 

It  is  frequently  desirable  to  show  two  or  more  ||  sees,  in  the  same  sectional 
view.  See  Fig.  190. 

A  sec.  may  sometimes  be  shown  as  revolved  upon  the  part  sectioned.  See 
sec.  on  D-D,  Fig.  188(a),  also  185,  198.  In  such  case  the  original  view  is  drawn 


ESSENTIALS  OF   MECHANICAL   DRAFTING 


complete  and  the  sec.  in  full  or  dashed  lines.      It  is  generally  better  to  "break" 
the  view  and  show  the  sec.  in  the  space. 

Instead  of  making  a  separate  sec.  view,  an  invisible  section  may  be  indicated, 
as  in  Figs.  182,  208. 


(°)  (ty  L  r'*l  (c)  (d; 

FIG.  189 

99.  Broken  Views.  When  only  a  portion  of  an  object  is  required  to  be 
shown  and  the  portion  is  not  limited  by  C.  Ls.,  the  views  may  be  broken,  as 
indicated  in  Figs.  191,  179,  180,  181,  203,  204.  The  outline  of  the  break  is  some- 
times omitted.  Fig.  188(a). 

The  broken  ends  of  symmetrical  pieces  may  often  be  made  to  suggest  the 
sectional  shape. 


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d 


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FIG. 190 


FLAT   BAR  WOOD 

FIG.  191 


WORKING   DRAWINGS 


100.     Standard  Sizes  of  Sheets  and  Scale  of  Drawings. 

(a)  For  convenience  in  handling,  filing,  etc.,  the  drawings  are  usually  made 
upon  sheets  cut  to  standard  sizes,  which  are  peculiar  to  each  shop  or  office  and 
dependent  largely  upon  the  uses  for  which  the  drawings  are  intended.  For 
sizes  in  common  use,  see  Art.  4. 


HALF  PL/|N 

Showing   Detail^  at  Top. 


M.mlw  .  Tfe         A       — \ 

= 


PEDESTAL 

Scale:    3"=  I  ft. 


FIG.  1 92 

(b)  All  drawings  are  made  to  a  definite  scale.  Art.  11.  The  scale  chosen 
must  be  such  that  all  parts  and  dimensions  will  be  shown  clearly.  Details 
should  be  to  full  size,  if  practicable;  small  details  to  an  enlarged  scale,  if  nec- 
essary. It  is  desirable,  however,  that  all  drawings  on  a  sheet  be  to  the  same 
scale. 


100 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


sze. 
Half 


Scales  commonly  used  are  full  size,  and  $,  J,  J,  fa  -^,  ^,  and  Jj 
These  are  usually  stated  on  the  drawing  thus:  Scale:  Full  Size.  Scale: 
Size.  Scale:  3"  =  1  ft.  Scale:  1$"  =  1  ft.,  etc. 

(c)  To  determine  the  scale  necessary  to  be  used  for  a  given  size  of  sheet, 
find  the  ratio  of  the  available  space  to  the  full  size  dimensions,  making  due  allow- 
ance for  spaces  between  the  views  and  from  margins.  Thus,  supposing  the 
drawing  space  horizontally  to  be  13£"  minus  2"  for  spaces,  and  the  hor.  di- 
mensions 20"  plus  85",  the  ratio  would  be  11^  to  23  \  and  the  nearest  convenient 


Top  removed 


TABORET 

Scale:      3"=  I  ft. 


FlG.  193 


scale,  considering  the  hor.  dimensions  and  spaces  only,  would,  therefore,  be  half 
size  or  6"  =  1  ft.  To  aid  in  such  calculations,  the  layout  sketch  would  be 
used.  Art.  105(a). 

101.  Dimensioning.  Although  the  drawing  is  made  to  a  definite  stated 
scale,  generally,  the  accuracy  of  the  views  themselves  is  not  depended  upon 
to  indicate  the  size  of  the  object  even  when  drawn  full  size;  all  dimensions 
required  by  the  workman  must  be  shown  upon  the  drawing  in  figures,  or  other- 
wise definitely  specified. 

In  order  to  give  the  essential  dimensions  and  to  omit  those  that  would  be 
unnecessary,  misleading,  or  impractical,  the  draftsman  must  consider  the  needs 
and  convenience  of  the  workman,  and  the  successive  steps  and  processes  involved 


WORKING   DRAWINGS 


101 


(b) 


FIG.  194 


in  the  construction  of  each  part.  In  addition  to  a  knowledge  of  shop  require- 
ments, he  must  exercise  good  judgment  in  deciding  where  to  place  the  dimensions, 
so  that  they  can  be  easily  found  and  applied.  This  article  describes  the  methods 
of  dimensioning  for  commonly  occurring  cases. 

(a)  FORMS  OF  DIMENSIONS.  To  allow  for  the  use  of  a  two-foot  rule  in 
working,  ordinary  dimensions  less  than  two  feet  are  usually  given  in  inches 
and  halves,  4th,  8ths,  etc. ;  thus,  17^".  Others  are  given  in  feet  and  inches;  thus, 
2'-0",  2  '-6",  2'-Oi";  or  thus,  2  ft.  0",  2  ft.  6",  2  ft.  0£".  Some  offices  use  inches 


FIG.  195 


102 


ESSENTIALS  OF  MECHANICAL  DRAFTING 


up  to  36,  others  up  to  72,  and  in  some  classes  of  work  all  dimensions  are  given 
in  inches.  When  the  greatest  possible  exactness  is  required,  dimensions  must 
be  given  in  decimal  form;  thus,  0".94,  2".442.  When  all  dimensions  are  under- 
stood to  be  in  inches,  the  sign  (")  may  be  omitted.  Limits  of  allowable  variation 
in  size  are  often  indicated  thus,  |:IH,  which  means  that  the  measurement 
must  not  be  greater  than  1.500  nor  less  than  1.498. 


(a) 


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Taper 

Z  Per  ft. 

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FIG.  196 

(b)  DIMENSION  LINES,  FIGURES,  AND  ARROWHEADS.  The  figures  are  placed 
upon  dimension  lines.  These  lines,  with  the  exception  of  those  for  radial  dimen- 
sions and  unlimited  distances,  have  arrowheads  at  both  ends.  For  forms  and 
sizes  of  figures  and  arrowheads,  see  Figs.  73,  74.  Arrowheads  must  touch  the 
lines  between  which  the  dimension  is  given,  and  the  figure  must  state  the  full  size 
of  the  corresponding  measurement  of  the  object  regardless  of  the  scale  of  the 
drawing.  Acceptable  methods  of  placing  figures  and  arrowheads  are  shown  in 
Figs.  196,  197. 


WORKING  DRAWINGS 


103 


Figures  should  be  so  placed  that  they  can  easily  be  read  from  lower  and 
right  sides  of  the'  drawing.  Avoid  placing  a  figure  where  it  will  interfere  with 
others,  or  with  other  lines. 

It  is  not  permissible  to  place  the  figures  for  more  than  one  dimension  between 
the  same  arrowheads,  nor  upon  other  than  dimension  lines. 


(a) 


FIG.  197 


(c)  LOCATIONS  OF  DIMENSIONS.  Dimension  lines  for  linear  measurements 
must  always  be  _L  to  the  parallels  between  which  the  dimensions  are  given,  and 
ordinarily  not  nearer  than  \"  to  object  lines  and  other  dimension  lines,  C.  Ls., 
etc.  They  should  not  cross  each  other  or  any  line  of  the  views,  nor  be  drawn 
as  continuations  of  other  lines,  if  avoidable.  In  general,  place  dimensions 
outside  of  views,  unless  greater  clearness  and  ease  in  reading  will  result  by  placing 


104  ESSENTIALS  OF  MECHANICAL  DRAFTING 

them  otherwise.  Dimensions  may  be  extended  beyond  parts  dimensioned  by 
means  of  extension  lines.  Pointers  may  also  be  used.  Figs.  196(a),  201. 

Never  dimension  a  distance  in  a  view  in  which  it  is  foreshortened.  Dimen- 
sion a  detail  preferably  in  a  view  in  which  it  is  visible.  So  far  as  practicable 
give  related  dimensions,  as  of  the  length,  width,  and  location  of  a  hole,  in  the 
same  view.  Dimensions  clearly  given  in  one  view  should  not  be  repeated  in 
another  upon  the  same  sheet.  When  parts  are  obviously  alike,  dimension  one 
or  two  only. 

Distances  should  be  given  from  lines  which  represent  finished  or  trued 
surfaces,  and  from  C.  Ls.  Do  not  give  distances  from  C.  Ls.  when  not  necessary. 
Locate  symmetrical  parts,  in  general,  by  giving  distances  to  their  C.  Ls.  Figs. 
196,  197,  183,  184.  In  a  series  of  such  parts,  give  distances  between  centers. 
As  a  rule,  give  dimensions  of  successive  distances  in  the  same  direction  from 
the  same  surface  or  C.  L.  The  final  dimension  of  a  series  is  sometimes  omitted 
to  indicate  this  surface  or  line  more  definitely.  Give  total  or  overall  dimensions 
as  well  as  all  detail  or  sub-dimensions,  so  that  the  workman  will  not  be  obliged 
to  calculate. 

Place  ||  dimension  lines  in  the  order  of  their  length,  the  longest  farthest 
from  the  part  dimensioned,  to  avoid  crossing  the  dimension  lines  and  causing 
confusion.  Figs.  180,  196,  198. 

(d)  ANGLES  AND  TAPERS.     Lines  appearing  to  be  hor.,  vert.,  or  J_  to  each 
other  are  assumed  to  be  so  unless  otherwise  dimensioned.      An    Z    may  be 
dimensioned  by  co-ordinate  dimension  lines,  or  by  an  arc  described  from  its 
vertex  as  center.      Fig.  196(c).      See  also  171  (a).      Common  methods  of  dimen- 
sioning tapers  are  shown  in  Fig.  196 (d).      Short  tapers  are  sometimes  dimensioned 
in  degrees.     Standard  tapers  are  specified  by  number  and  kind;  as,  "No.  2  Morse 
Taper,"  etc. 

(e)  CIRCLES  AND  ARCS.     In  general,  give  the  diam.  dimension  of  a  O   and 
place  the  figure  on  an  oblique  diam.,  or  extend  the  dimension  to  the  most  con- 
venient location.    Fig.  197 (a).      Small  Os  may  be  dimensioned  as  in  Fig.   (b). 
Give  the  rad.  of  an  arc  with  an  arrowhead  at    the    curve    end   only.      When 
the  space  is  too  small  for  the  figure,  dimension  as  in   (c).      Radii  \"  or  less, 
of  fillets  and  finish  arcs,  ordinarily  need  not  be  given.     When  the  rad.  is  known 
or  unimportant,  indicate  by  Rad.  or  R.     When  holes  or  bolts  are  arranged  in  a 
O,  give  diam.  or  rad.  of  the  circular  C.  L.     When  equal  spacing  is  not  evident, 
specify  by  note  "Equally  Spaced."     If  unequal,  dimension  as  indicated  in  Fig. 
197(b). 

(f)  IRREGULAR  OR  NON-CIRCULAR  CURVES.     These  may  usually  be  dimen- 
sioned by  giving  the  lengths  and  positions  of  offsets  _L  to  appropriate  base  lines. 
Fig.  198,  also  174(a).     In  most  cases,  however,  it  is  more  practical  to  omit  such 
dimensions*,  and  to  provide  the  workman  with  one  or  more  exact  patterns  or 
templets  of  thin  material  against  which  he  can  lay  out  the  curves  directly  on 
the  piece. 


*Fig.  198  is  thus  dimensioned  merely  to  illustrate  the  method. 


WORKING   DRAWINGS   . 


105 


(g)  ROUND,  SQUARE,  HEXAGONAL,  AND  OCTAGONAL  PIECES.  When  the  cir- 
cular, square,  etc.,  shape  is  not  shown  in  any  view,  nor  specified  by  note  or  title, 
indicate  by  Dia.  or  D.,  Sq.,  Hex.,  or  Oct.,  after  the  diameter  dimension.  Figs. 
174(a),  199. 

(h)  STANDARD.    MEASUREMENTS. 

Screws,   pipes,   bolts,   and  springs:     give  dimensions  as  in  Chap.  XI. 

Tubing:  give  outside  diam.  and  thickness  by  gage,  or  in  thousandths  of 
an  inch. 

Wire:     give  diam.  by  gage,  or  in  thousandths  of  an  inch. 

Sheet  Metal:     give  thickness  by  gage,  or  in  thousandths  of  an  inch. 

(i)  ASSEMBLY  DRAWINGS.  The  dimensions  required  depend  largely  upon 
the  kind  of  object  and  the  purpose  of  the  drawing.  In  machine  assemblies 
usually  only  the  important  overall  dimensions  and  locations  of  principal  C.  Is. 
are  necessary. 


FIG    198 


I3f— -| 


102.     Lettering. 

(a)  EXPLANATORY  NOTES,  ETC.  Specifications  and  directions  concerning  the 
kind  of  material  to  be  used,  the  name  and  number  wanted  of  each  piece,  kind 
of  finish  to  be  given,  kind  of  fit  required,  and  any  other  necessary  information 
not  shown  by  the  drawing  or  stated  in  a  title  or  bill  of  material,  must  be  expressed 
in  brief,  concise  notes  lettered  upon  the  sheet.  See  Art.  25.  Notes  should 
preferably  be  located  outside  of  the  views  and  to  read  horizontally,  or  vertically 
from  the  bottom.  The  part  noted  may  be  indicated  by  a  pointer.  Fig.  188. 

Finished  Surfaces.  When  a  surface  of  a  casting  or  forging  is  to  be  machined 
or  finished  by  filing,  turning,  milling,  grinding,  etc.,  an  allowance  must  be  made 
on  the  piece  for  this  finish.  Ordinary  finish  is  generally  indicated  by  a  letter  f, 
placed  on  the  surface  in  all  views  which  show  the  surface  as  a  line.  Fig.  196,  also 


106 


ESSENTIALS  OF  MECHANICAL   DRAl-TIMi 


189  (a),  (b).  The  f  s  should  be  placed  to  read  horizontally  as  shown.  When  the 
picco  is  to  be  finished  all  over  the  f's  are  omitted,  and  the  note  "Finish  all  over" 
or  "F.  A.  O."  placed  near  the  principal  view.  The  limits  of  a  finished '  portion 


Top  Removed 


SECTION     AT  A-B. 


Q.S.WOak. 


JARDINIERE   STAND 

Scale:     3"=  I  ft 


FIG.  199 

may  be  indicated  as  in  Fig.  196(a),  or  by  a  note.  It  is  often  necessary  to  specify 
the  kind  of  finish;  as  "Polished,"  "Ground,"  "Milled,"  "Reamed,"  etc.  In  this 
case  the  f's  are  also  usually  omitted. 

When  a  portion  of  an  unfinished  surface  is  to  be  machined  to  form  a  bearing 


WORKING   DRAWINGS 


107 


FlG.  200 


for  a  bolt-head,  nut,  stud-shoulder,  etc.,  it  may  be  specified 
thus,  "Spot  Face, "or  "Counterbore  to  Surface,"  according  as  an 
allowance  is,  or  is  not,  to  be  provided  for  this  finish. 

Knurled  surfaces  are  indicated  by  the  word  "Knurl,"  or  as 
in  Fig.  200.  The  varying  spaces  and  Z  s  of  the  lines  are  esti- 
mated by  eye. 

Circular  Holes.      "Cored,"  "Drilled,"   "Countersunk,"  etc., 

holes  should  be  specified  in  one  view,  or  the  other,  as  in  Fig.  201;  "Tapped"  holes 
as  in  Fig.  218. 

Kind  of  Fit.     When  one  metal  piece  is  to  be  fitted  into  another,  the  fit  is 
specified  as  "Running,  "Drive,"  "Force," 
or  "Shrink"   Fit;    or    by    indicating    the 
limits  of  variation  in  the  sizes  of  the  parts. 
See  Art.  101  (a). 

Treatment  of  Metal.  Special  treatment 
is  specified,  as  "Tempered,"  "Hardened," 
"Case  Hardened,"  "Blued,"  etc. 

Standard  Parts.  Parts  such  as  screws, 
bolts,  keys,  etc.,  which  conform  in  design 
and  dimension  to  recognized  commercial 
standards  are  usually  omitted  from  the 
detail  drawings  and  specified  by  note,  or 
listed  in  a  bill  of  material.  Art.  (d). 

(b)  IDENTIFYING    MARKS.     It'  is  cus- 
tomary to  give  each  piece  of  a  machine 
a  distinguishing  number  or  letter  under 
which  it  is  listed  and  referred  to.     A  cor- 
responding mark  is  placed  on  the  drawing 
of  the  piece  and  is  often  accompanied  by 
its  name.     Figs.  202-206,  210-213. 

(c)  TITLES.     For  convenience  in  filing, 
etc.,  the  title  of  the  drawing  is  generally 
placed  in  the  lower  right  corner.     Figs. 
179-180,   202-206.     The  title  should  des- 
ignate:— 

1.  The  name  of  the  object,  part,  or 
particular  detail  shown,  or  all  three. 

2.  The  scale,  if  uniform. 

3.  The    date    of    completion    of    the 
drawing. 

4.  The  draftsman's  signature. 

If  a  single  detail  only  is  shown,  the 
number  required  and  material  are  gener- 
ally stated.  Some  or  all  of  the  following 
information  is  also  generally  included: 
The  type  of  drawing, — as  assembly,  detail,  Fi G.  20 1 


n 

|      L4^t 

/Drill ,  ^Counter bore. 


Dri  IN 


108  ESSENTIALS  OF  MECHANICAL   DRAFTI.NC 

shop,  etc.;  the  signatures  of  the  tracer  and  persons  by  whom  the  drawing  is 
c -In-eked  and  approved;  the  firm  for  whom  the  drawing  is  made;  references  to 
other  drawings,  etc. 

Sub-titles.  On  a  sheet  of  details  each  piece  may  have  a  title,  consisting  of 
its  name  and  identifying  mark;  the  scale,  if  not  stated  in  the  main  title;  number 
required  if  more  than  one;  material;  pattern  number,  if  a  casting;  and  finish, 
if  to  be  all  over.  If  a  bill  of  material  is  given,  the  identifying  mark  and  the 
scale  only  are  necessary. 

Planning  a  Title.  First  write  the  title  upon  a  separate  paper.  Having 
decided  upon  the  statements  to  be  included,  and  the  size  and  style  of  lettering, 
proceed  to  draw  the  vert.  C.  L.  of  the  title  space  and  the  guide  lines  for  the 
heights.  Next,  upon  another  paper  and  adjacent  to  its  upper  edge,  compose 
and  sketch  the  first  line  of  lettering  to  the  size  to  be  used  upon  the  drawing. 
Then,  placing  this  sketch  against  the  proper  guide  line,  with  the  middle  point 
of  the  line  of  letters  at  the  C.  L.,  point  off  the  widths  of  the  letters  and  draw 
each  carefully.  Proceed  in  like  manner  with  other  lines  of  lettering. 

(d)  FILING  INDEX  AND  BILL  OF  MATERIAL.     A  filing  index  giving  the  num- 
ber or  other  designating  mark  of  the  machine  and  the  sheet  number  is  usually 
placed  in  the  lower  right  corner  or  included  in  the  main  title.     It  is  often  shown 
in  an  upper  corner  also.     Figs.  202-206,  210-213. 

In  a  set  of  drawings,  the  assembly  may  be  indexed  as  sheet  1,  and  may  include 
a  list  of  the  other  drawings  with  their  numbers.  The  sheet  number  of  each 
detail  may  be  given  near  its  identifying  mark;  thus,  (3-5):  the  3  indicating  the 
mark,  and  the  5  the  sheet  no.  containing  the  detail. 

A  sheet  of  details  is  usually  accompanied  by  a  bill  of  material  placed  above 
or  at  the  left  of  the  main  title,  accounting  for  each  piece  and  giving  its  mark, 
name,  material,  no.  wanted,  and  other  necessary  description  and  information. 
For  material  to  be  cut  to  size,  the  rough  stock  dimensions  should  be  specified. 

When  the  number  of  parts  is  large  or  more  than  one  sheet  of  details  is  neces- 
sary, a  separate  bill  grouping  the  forgings,  castings,  etc.,  and  giving  the  sheet 
no.  of  each  part,  may  be  made.  Fig.  213. 

(e)  ABBREVIATIONS.     The  following  are  some  of  those  in  common  use: — 
R.  H.  Right-hand      S.  Steel  O.  H.  S.  Open  Hearth  Steel 
L.  it.   Left-hand         M.  S.       Machine  Steel  C.  H.  S.  Case  Hardened  Steel 
C.  I.     Cast  Iron         T.  S.        Tool  Steel                   S.  C.         Steel  Casting 

W.I.    Wrought  Iron    C.  R.  S.  Cold  Rolled  Steel      Bz.  Bronze 

103.  Shadow  Lining  and  Line  Shading.     Arts.  23,  24.     Shadow  lines  and  line 
shading  are  used  when  the  advantage  gained  in  clearness  and  effect  is  sufficient 
to  warrant  the  expenditure  of  the  time  necessary  to  apply  them,  otherwise  they 
are  omitted.     They  are  of  special  value  on  drawings  of  complicated  objects 
shown  by  few  views  or  whose  corresponding  views  are  upon  different  sheets, 
as  in  some  assembly  drawings.      They  are    rarely  used  on  ordinary  detail  or 
shop  drawings. 

104.  Sketching.     To  the  designer,  draftsman,  or  mechanic,  skill  in  making 
freehand  sketches  for  the  rapid  expression  of  ideas  of  exact  form  and  structure, 


-oi-ns 


FIG    2O2 


i~ULj  "ULu") 


IOIM      SPEED      LATHE 
ASSEMBLY 

DAY     MFG      CO        BOSTON 
Scota    6'-IFt  Date        \~29-\G 

Dr  ^HrT     Tr  *?&£.      Cb.7X.f~-    ADO.  £7 


SL-IO- 


WORKING   DRAWINGS  109 

and  as  an  aid  in  the  solution  of  constructive  problems,  is  of  great  importance. 
To  the  student,  careful  sketching  is  fully  as  valuable,  as  a  means  of  acquiring 
ability  to  make  and  to  read  working  drawings,  as  instrumental  drawing. 

(a)  In  making  working  sketches  from  objects  the  following  order  should  be 
observed  :— 

1.  Separate  the  parts,   if  necessary,   and  sketch  the  views  of  each  in  detail. 
Begin  with  the  main  C.  Ls.  and  principal  surfaces,  blocking  in  first  the  larger 
details  of  the  piece  and  proceeding  in  like  manner  down  to  the  minor  details. 
See  Fig.  209  (a),  (b),  (c). 

Distances  and  directions  are  generally  determined  by  eye.  Co-ordinate 
paper,  ruled  in  \"  or  i"  squares,  is  often  used  and  affords  a  more  ready  and 
accurate  means  of  obtaining  the  desired  proportions,  etc.  For  use  of  pencil 
see  Art.  9(b). 

The  number  and  character  of  the  views  should  be  such  as  to  express  all 
required  facts  clearly.  Leave  nothing  to  memory.  Make  the  sketches  large 
enough  to  prevent  crowding  the  notes  and  figures.  The  sketches  should  be 
intelligible  to  any  one  familiar  with  working  drawings,  and  should  enable  the 
scale  drawings  to  be  readily  made  from  them  without  having  to  resort  to  the 
object.  Isometric  or  oblique  views  may  often  be  used  to  advantage. 

2.  Indicate  the  finished  surfaces  and  put  on  the  necessary  extension  lines, 
dimension  lines,  and  arrowheads. 

3.  Obtain  the  dimensions   by  careful   measurement  of  the  object    and  put 
on  the  explanatory  notes. 

Each  piece  should  be  dimensioned  independently  of  the  others. 

(b)  MEASURING   OBJECTS.     In   taking   measurements,    a  foot,  or  a  two-foot 
rule  may  be  used  for  ordinary  work  and  steel  rules,  gages,  etc.,  for  fine  work. 
Obtain  distances  from  trued  or  finished  surfaces  whenever  possible.     In  obtain- 
ing inside  and  outside  diams.,  calipers  may  be  used. 

In  an  object  of  varying  diams.  (Fig.  174 (a)),  measure  the  diams.  at  a  sufficient 
number  of  pts.  and  locate  these  diams.  by  measurements  ||  to  the  axes.  In 
measuring  an  irregular  form,  as  the  table  leg  (Fig.  198),  it  is  necessary  to  es- 
tablish a  base  line,  as  A-B,  by  means  of  a  triangle  or  a  carpenter's  square, 
and  to  determine  the  lengths  of  _]_  offsets  to  it  with  a  foot  rule 

In  locating  a  circular  hole,  measure  to  edge  of  hole  and  add  half  its  diam. 

In  ordinary  finished  surfaces,  take  the  nearest  32d;  in  rough  work,  the 
nearest  16th;  in  finely  finished  objects  and  parts,  absolute  exactness  is  necessary. 

105.     Making  Scale  Drawings. 

(a)  Having  completed  the  sketches,  decide  upon  the  number  and  arrange- 
ment of  the  views  to  be  shown  in  the  drawing,  and  the  necessary  scale.  To 
aid  in  determining  the  scale  and  the  locations  of  the  chosen  views  upon  the 
sheet,  a  rough  layout  sketch  indicating  the  general  outlines,  main  C.  Ls.,  and 
margins,  should  be  made.  Thus,  from  the  layout  (Fig.  207)  for  the  bearing 
shown  in  Fig.  208,  it  will  be  seen  that  the  minimum  space  required  horizontally, 
not  including  spaces  between  views  and  from  margins,  will  be  2a  +  2b.  Simi- 
larly, that  required  vertically  will  be  c  +  d  +  2b.  From  these  two  sets  of 


c 


UE 


aOJ.     1HL    JO 


-nv  -  IION 


V 


'Hi!,  ^4-- 
±-     :  J.± 


b^5M 


u 


7  & 


*  9 
>:  ^ 


FIG.  203 


FIG.  204 


FIG.  205 


FIG.  206 


114 


ESSENTIALS  OF  MECHANICAL   DRAFTING 


*       1 — — 

rtf-H-4-  . 


b-J 


FIG. 207 


j.     |    I, Core.. |  Deep 


Cast  Iron    Pa+t  *  B-5 


PLAIN    BEARING 

Scale:  Full  Sirt       Date'  3-7-18 
Dr.  N€..  Tr  M..    Ch.  AP.    App  CF. 

M-I8-I 


Fio.208 


WORKING  DRAWINGS 


115 


dimensions,  with  due  allowance  for  spaces  and  title,  the  scale  may  be  determined, 
as  in  Art.  100(c).  Having  decided  this,  the  dimensions  A,  B,  C,  and  D  for 
the  locations  of  the  C.  Ls.,  should  be  estimated,  proper  allowance  being  made 
where  necessary  to  preserve  a  well-balanced  sheet. 

The  same  general  directions  should  be  observed  in  all  drawings,  as  lack  of 
provisions  for  the  necessary  views,  etc.,  may  lead  to  errors  impossible  to  rectify 
without  re-drawing  the  entire  sheet. 

(b)  Having  completed  the  layout,  the  scale  drawing  (Fig.  208)  may  be  begun. 
Four  stages  in  penciling  are  indicated  in  Fig.  209.  For  general  working  instruc- 
tions, see  Art.  3.  In  drawing  from  objects,  the  detail  drawings  are  first  made 
and  the  general  drawing  built  up,  piece  by  piece,  from  them.  In  designing 


FOUR 


STAGES   IN   PENCILING 
FIG  209 


or  planning  objects  to  be  built,  the  general  drawing  is  usually  first  begun  and 
the  detail  drawings  worked  out  from  it.  In  some  cases  it  is  necessary  to  carry 
along  both  detail  and  general  drawings  at  the  same  time. 

106.     Tracing  and  Blue-printing. 

(a)  Instead  of  inking  the  pencil  drawing,  tne  finished  drawing  is  usually 
obtained  by  tracing  in  ink  upon  tracing  linen  or  paper  fastened  over  the  original. 
This  tracing  is  then  filed  as  the  permanent  record  of  the  construction  and  used 
for  making  blue-prints  or  other  copies  for  shop  and  general  use. 

The  original  is  sometimes  penciled  directly  upon  the  tracing  material. 


0> 


FIG.  210 


FIG   211 


514 


FIG. 212 


FiG.213 


120  ESSENTIALS  OF   MECHANICAL   DRAFTING 

The  dull  side  of  tracing  linen  takes  the  ink  better,  but  erasures  can  be  made 
more  readily  upon  the  glazed  side.  Either  will  take  the  ink  more  readily  if 
rubbed  with  a  cloth  and  chalk  powder.  Penciling  should  be  done  on  the  dull 
side.  Carry  but  little  ink  in  the  pen,  and  make  the  lines  somewhat  wider  than 
ordinary  as  the  lines  print  finer  than  those  of  the  tracing. 

As  the  linen  contracts  and  expands  unevenly,  tracings  that  cannot  be  com- 
pleted the  same  day  should  be  inked  by  sections. 

Make  ink  erasures  with  hard  eraser,  rubbing  gently  to  avoid  injury  to  the 
surface.  Restore  the  smoothness  by  rubbing  with  soapstone  or  a  smooth  piece 
of  bone.  Pencil  lines  may  be  erased  with  soft  eraser. 

(b)  Blue-prints  are  usually  taken  in  a  printing  frame  having  a  glass  front 
and  removable  backboard.  The  tracing,  or  original  drawing  if  on  translucent 
material,  is  placed  with  the  drawing  side  next  to  the  glass  and  its  under  side 
in  contact  with  the  chemical  coated  surface  of  blue-print  paper.  The  glazed 
side  of  the  frame  is  then  exposed  to  direct  sunlight,  which,  penetrating  the  part 
of  the  tracing  material  not  covered  by  the  ink  lines,  causes  the  chemical  not 
thus  protected  to  change  color  and  to  adhere  permanently  to  the  paper,  while 
that  under  the  lines  remains  unchanged.  After  suitable  exposure  the  paper  is 
removed  from  the  frame  and  soaked  in  water,  which  dissolves  the  unfixed 
chemical,  leaving  white  lines  on  a  blue  ground. 

107.  Checking  Drawings.     It  is  customary  not  to  permit  a  drawing  to  be 
worked  from  until  it  has  been  checked  by  a  careful,  systematic  examination, 
and  approved  by  the  head  draftsman.     In  checking,  it  is  well  to  assume  every- 
thing to  be  incorrect  until  proved  to  the  contrary.     The  following  order  may 
be  observed: — 

See  that  each  piece  has  been  represented  and  that  its  views  are  properly  related. 

Check  views  of  each  piece  for  correct  and  adequate  description  of  form  and 
construction. 

Note  if  C.  Ls.  and  all  necessary  dimensions  and  notes  are  given. 

Scale  every  dimension,  and  verify  overall  dimensions  by  computation. 

Compare  the  figures  on  all  parts  that  are  to  fit  together. 

Check  measurements  in  details  and  assembly  and  note  if  they  agree. 

Finally,  see  that  all  items  required  to  be  recorded  in  the  title  and  bill  of 
material  are  complete  and  correct. 

108.  Reading  Drawings.     Ability   to   read   working   drawings   rapidly   and 
intelligently  is  quite  as  important  as  skill  in  making  them.     This  ability  can 
be  acquired  through  the  study  of  such  drawings  and  comparison  with  the  objects 
represented;    through   the  execution   of   drawings   from   objects;    through   the 
making  of  mechanical  pictorial  drawings  and  developments,  from  good  examples; 
and  by  making  the  objects,  that  is,  working  from  drawings. 

In  reading  a  drawing,  first  fix  in  mind  the  general  shape  of  the  main  body 
of  the  object,  observing  if  the  outline  shows  it  to  be  rectangular,  cylindric,  etc., 
or  a  modification  of  such  forms.  Then  observe  modifications  of  the  general 
shape,  proceeding  from  the  more  important  details  down  to  the  minor  details. 

Note  carefully  the  conventional  methods  employed  in  the  representation 
and  complete  mentally  the  graphic  statement  of  what  is  required. 


WORKING   DRAWINGS  121 

Endeavor  to  visualize  the  object;  to  see  in  each  view,  not  mere  lines, 
but  the  object  itself  as  if  standing  out  of  the  paper.  Regard  the  front  view 
as  the  object  directly  in  front  of  you;  in  looking  at  the  top  view  imagine  your- 
self looking  down  upon  the  object;  in  looking  at  the  side  view  imagine  yourself 
as  viewing  the  object  in  a  direction  at  right  Zs  to  the  front,  and  so  on. 

Finally,  note  the  dimensions,  and  specifications  as  to  materials,  finish,  etc. 
All  information  as  to  sizes  should  be  obtained  from  the  figured  dimensions  and 
specifications;  rarely  by  measuring  the  drawing  itself. 

It  is  evident  that  full  knowledge  of  the  form,  size,  and  relation  of  the  lines 
and  surfaces  of  each  part  of  the  object  represented  can  be  secured  only  through 
the  information  shown  in  all  the  views  taken  together. 


CHAPTER  XI 


HELICAL    CURVES,    THREADED    PARTS,    AND    SPRINGS 


109.  Helices.  If  a  pt.,  A  (Fig.  214)  be  imagined  to  move  along  the  generating 
line,  A-12,  of  a  surface  of  revolution,  while  the  line  itself  revolves  about  the  axis, 
the  pt.  will  generate  a  line  of  double  curvature  called  a  helix. 

The  distance  that  the  pt.  advances,  measured  ||  to  the  axis,  during  one 
revolution  of  the  line,  is  called  the  pitch  of  the  helix. 

If  the  rate  of  motion  of  the  pt.  and  generating  line  of  a  cylindric  helix  be 
uniform,  that  is,  if  the  pt.  advances  j,  \,  or  other  fractional  part  of  the  pitch 
distance  during  the  same  fractional  part  of  a  turn,  the  helix  is  uniform  or 
equable;  if  otherwise,  it  is  variable. 

Again  the  curve  is  a  right-hand  or  left-hand  helix,  according  as  the  generating 
pt.  rises  to  the  right  or  to  the  left  in  the  front  half  of  a  turn  when  the  axis  is  vert. 

The  helix  has  many  applications  in  mechanical  and  architectural  design, 
notably  in  screw  threads,  some  forms  of  springs,  winding  stair  rails,  etc. 

(a)    To  DRAW  A  RIGHT-HAND    EQUABLE    CYLINDRIC    HELIX.       Fig.  214.       Assume 

and  represent  equidistant  positions  of  the  generating  element,  as  1,  2,  3,  etc. 
Divide  the  pitch  distance,  as  A-A12,  into  the  same  number  of  equal  parts  and 
draw  hors.  through  the  pts.  of  division  to  cut  the  elements  at  A1,  A2,  A3,  etc. 
Assuming  A  to  be  the  generating  pt.,  A1,  A2,  A3,  etc.,  will  be  twelve  pts.  of  the 
desired  curve,  for  evidently  when  the  generating  ele- 
ment has  made  TT  of  a  revolution  the  advancing  pt. 
A  will  have  moved  TJ  of  the  pitch  distance  and  will, 
therefore,  be  in  the  element  1  at  A1.  When  the  element 
has  moved  ^  a  revolution  the  pt.  A  will  have  moved 
$  of  the  pitch  distance  and  will  be  in  the  element  6 
at  A6,  and  so  on. 

As  the  curvature  is  more  abrupt  at  the  outer 
elements,  additional  pts.  should  be  determined  by 
sub-division  as  shown.  Having  fixed  the  pts.  with 
the  needle,  trace  the  helix  through  them  freehand. 

FIG. 214 
DEVELOPMENT  OF  HELIX 


CYLINDRIJC    HELIX 


122 


HELICAL   CURVES,   THREADED   PARTS,   AND   SPRINGS  123 

Points  for  other  turns  of  the  curve  may  be  located  by  stepping  off  the  pitch 
distance  upon  the  elements,  from  the  pts.  of  the  first  turn.  The  manner  of 
obtaining  parallel  or  double  helices  is  evident;  also  the  method  of  drawing 
conic  or  other  helices. 

(b)  In  ruling  and  inking  the  helix  observe  directions  given  in  Art.   18(a). 
When  several  cylindric  helices  of  the  same  pitch  are  to  be  drawn,  a  templet 
of  a  half  turn,  as  A- A6,  may  be  made. 

(c)  The  development  of  a  cylindric  helix  of  one  turn  is  the  hypotenuse  of 
a  right  A  whose  base  is  equal  to  the  circumference  and  whose  altitude  equals 
the  pitch,  as  shown. 

(d)  Any  desired  motion  of  a  pt.  may  be  plotted  in  a  development  of  a  cylinder 
and  then  projected  back  to  the  view,  as  for  example  in  designing  a  cam  for 
converting  circular  into  reciprocating  motion. 

110.  Screw  Threads.  If  a  cylindric  bar  be  revolved  at  a  uniform  velocity 
upon  its  axis,  in  a  lathe,  and  the  point  of  a  V-shaped  cutting  tool  be  pressed 
against  its  surface  and  moved  at  a  uniform  rate  parallel  to  the  axis,  the  tool 
will  cut  a  helical  groove,  V-shaped  in  section.  If  this  groove  be  cut  so  that 
a  similar  projecting  portion  is  left  upon  the  bar,  a  V  screw  thread  will  be  formed. 
Fig.  215(a). 

Similarly,  if  a  square  groove  be  cut  and  a  square  projecting  portion  left  upon 
the  bar,  a  square  screw  thread  will  be  formed.  Fig.  215(c). 

Fig.  (b)  illustrates  a  sec.  of  a  V-threaded  and  Fig.  (d)  of  a  square-threaded 
hole.  Observe  that  the  helical  lines  of  the  threads  correspond  to  the  invisible 
lines  of  the  screws. 

The  V  thread  is  commonly  used  on  screws  for  fastening  purposes;  the  square 
thread  generally  on  screws  for  transmitting  motion  in  the  direction  of  the  axis. 
All  other  threads  are  modifications  of  the  V  and  square  forms. 

A  screw  thread  is  right-  or  left-hand  according  as  its  helices  are  right-  or  left- 
hand.  Thus  a  right-hand  screw  would  turn  around  to  the  right  (clockwise) 
in  advancing  or  entering  the  part  into  which  it  is  inserted. 

The  diameter  of  the  top  of  the  thread  is  called  the  nominal  diameter  of  the 
screw.  The  diam.  of  the  bottom  of  the  thread  is  the  root  diameter.  The  distance 
from  the  root  to  the  top  of  the  thread  measured  _l_  to  the  axis  is  the  depth  of 
the  thread.  The  distance  between  the  centers  of  adjacent  threads  measured 
||  to  the  axis  is  called  the  pitch  of  the  thread.  The  term  "pitch"  is  often  used 
to  designate  the  no.  of  threads  per  inch;  thus  "14  pitch"  means  14  threads  to 
the  inch.  The  distance  that  the  screw  would  advance  in  one  turn  is  called  the 
lead  of  the  screw.  In  a  single  thread  screw  the  lead  equals  the  pitch;  in  a  double 
or  triple  thread  it  is  two  or  three  times  the  pitch. 

(a)  MULTIPLE  THREADS.  Screws  are  generally  right-hand  and  single  thread 
as  shown  in  Fig.  215  (a),  (c).  If  the  pitch  of  a  screw  having  the  diam.  and  thread 
sec.  shown  in  Fig.  (c)  be  made  say  two  or  three  times  as  great,  the  increase  in 
the  depth  would  obviously  be  such  as  to  weaken  the  screw  at  the  root.  There- 


124 


1>SKNT1AI.S    ol     MIX'IIANK  AI.    DK.M-TINC 


V   SCREW  THREAD 


SQUARE  SCREW   THREAD 

Double 

(e) 


L=Lead  of  Scfew. 
P- Pitch  of  Thread 
d  -  Depth  of  TKd 


FIG  215 


d=  5P  *  01 

W=  3707   P  -  0052 


D«Nominal    Diam  of  Screw 
P«  Pitch  *•£ 

n=  No  of  Threads  per  inch 
d  =  Depth   of  Th'd 

FIG  216 


HELICAL  CURVES,   THREADED   PARTS,   AND   SPRINGS 


125 


fore,  in  designing  a  screw  of  which  the  lead  shall  be  two  or  three  times  the  pitch, 
instead  of  cutting  a  single  thread,  a  second  or  third  independent  parallel  thread 
would  be  cut.  Such  would  be  a  double,  or  triple  thread  screw. 

Fig.  (e)  represents  a  right-hand  double  square  thread  of  the  same  diam.  and 
pitch  as  the  single  thread  of  Fig.  (c). 

Observe  that  in  a  single  thread  screw,  the  top  on  one  side  is  diametrically 
opposite  the  bottom  on  the  other,  while  in  a  double  thread  the  tops  are  opposite. 

(b)  To  DRAW  A  SCREW  THREAD.      The   diam.,    pitch,  and  thread  sec.  being 
given,  as  in  Fig.  215(a);  first  obtain  the  elevation  and  an  end  view,  or  half  of 
revolved  base  of  a  cylinder  whose  diam.  equals  the  nominal  diam.  of  the  screw. 
Lay  off  the  pitch  A-C  and  draw  the  sec.  ADC.       Then  beginning  at  A,  draw 
the  helix  A-B-C-,  etc.,  for  the  top  of  the  thread  by  Art.  110(a).      To  obtain 
the  root  helix,   draw  a  semicircle  in  the  base  view  concentric  with-  the  first, 
obtaining  the  rad.  by  projecting  from  D.     Then  beginning  at  D  and  using  the 
same  pitch,  proceed  as  with  the  outer  helix.      Observe  that  the  outlines  of  the 
thread  come  outside  of  the  V  sec.  and  are  tangent  to  the  helices.      When  the 
pitch  is  small,  they  practically  coincide  with  the  sec.  outline. 

(c)  STANDARD    PROPORTIONS.     In   the   preceding   figures,   the   threads   were 
represented  with  large  pitch  in  order  to  show  the  construction  more  clearly. 
The  proportions  of  the  threads  most  commonly  used  and  the  formulae  for  obtain- 
ing them  are  given  in  Fig.  216.     In  this  country  the  Z  of  the  V  thread  is  usually 
60°,  but  for  general  work  the  tops  and  bottoms  are  flattened  as  shown  in  Fig.  (a). 

The  following  table  gives  the  U.  S.  St'd  no.  of  threads  per  inch,  for  diams. 
from  \"  to  4i". 

U.  S.  STANDARD  SCREW  THREADS 


DIAM. 

SCREW 

THDS. 
PER  IN. 

DIAM. 

SCREW 

THDS. 
PER  IN. 

DIAM. 

SCREW 

THDS. 
PER  IN. 

DIAM. 

SCREW 

THDS. 
PER  IN. 

DIAM. 

SCREW 

THDS. 
PER  IN. 

I 

20 

f 

11 

1 

8 

If 

5 

3 

3* 

T5* 

18 

H 

11 

1| 

7 

If 

5 

3J 

3| 

I 

16 

3 

10 

11 

7 

2 

41 

3^ 

31 

TV 

14 

if 

10 

If 

6 

2i 

*i 

3f 

3 

1 

13 

7 
8 

9  ' 

1| 

6 

2^ 

4 

4 

3 

tV 

12 

H 

9 

If 

% 

2f 

4 

4J 

21 

On  square  threads  the  no.  of  threads  per  inch  is  commonly  \  of  the  U.  S. 
St'd.  In  drawing,  the  depth  is  made  |. 

The  Acme  Standard  or  29°  thread  is  used  for  the  same  general  purpose  as 
the  square  thread.  Threads  per  inch  are  likewise  usually  the  same.  In  drawing, 
the  angle  is  made  30°. 

In  the  Whitworth  or  British  Standard,  threads  per  inch  are,  with  a  few  excep- 
tions, the  same  as  U.  S.  St'd.  See  handbooks. 


126 


I  581  NTI.M.S  OF   MECHANICAL   DRAFTING 


(d)  CONVENTIONAL  REPRESENTATION  OF  THREADS.  The  true  drawing  of 
the  thread  curves  involves  considerable  labor  and  in  small  screws  would  be 
impossible.  In  large  threads  it  is  customary  to  substitute  st.  lines  for  the  helices, 
as  in  Fig.  217.  In  invisible  threads  they  are  often  omitted  altogether. 

V  threads  less  than  one  inch  diam.  are  usually  represented  as  in  Fig.  218  (a), 
(b),  (c),  (d).  The  methods  (e),  (f),  (g),  (h)  are  also  used.  The  spacing  is 
estimated  by  eye,  without  regard  to  the  actual  no.  of  threads  per  inch,  but  in 
methods  (a),  (b),  (c),  (d)  the  positions  of  the  lines  should  indicate  whether 
the  screw  is  right-  or  left-hand.  The  thread  of  a  long  piece  may  be  shown  as 
in  (i). 


V   THREAD 


SQUARE  THREAD  ACME  THREAD        SINGLE   R.H.  V  TH'D   HOLE 


\r  ~7> 


Plbn 


=£= 

eva 


Doujbl 


Small  square  threads  are  usually  represented  as  in  (j).  The  exact  no. 
of  threads  per  inch  is  shown  unless  the  scale  is  very  small. 

A  threaded  hole  is  generally  represented  in  the  circular  view  as  in  (k), 
to  distinguish  it  from  a  drilled  hole;  a  drilled  and  threaded  hole,  as  in  (1). 

The  diam.  of  the  outer  O  in  each  is  equal  to  that  of  the  bolt  or  screw;  that 
of  the  smaller  about  equal  to  that  of  the  root.  Any  of  the  methods  of  Fig.  218 
may  be  used  for  the  other  views.  On  crowded  drawings  it  is  often  best  to  use 
methods  (d),  (f),  (h),  or  (m).  It  is  better  to  omit  the  drawing  of  the  threads 
beyond  the  screw  end  (see  (n)  ),  unless  method  (m)  is  used.  The  point  made 
by  the  drill  is  usually  shown. 

Figs,  (o)  and  (p)  show  methods  of  representing  small  pieces  in  sec.  when 
V-threaded  inside  and  outside. 

(e)  DIMENSIONING.  Give  the  outside  diam.;  the  no.  of  threads  per  inch, 
thus:  lOTh,  lOThds.,  10P.,  10,  or  X:  and  the  length  of  the  threaded  portion, 
from  the  end  when  chamfered,  and  from  the  curve  when  rounded.  If  the  thread 
is  other  than  right-hand  and  single,  specify  as  indicated  in  Figs,  (b),  (c). 


HELICAL   CURVES,   THREADED  PARTS,   AND  SPRINGS 


127 


00  (I) 

,10  Th.         |"Tap   10  Th 

II    2°Below  Surface. 


$  Pipe  Tap. 


13S  ESSENTIALS  OF  MECHANICAL  DRAFTING 

In  a  threaded  hole,  give  the  depth,  the  diam.  of  the  piece  to  be  screwed  into 
it,  and  no.  of  threads  per  inch.  Indicate  diam.  and  no.  of  threads  of  a  tapped 
hole,  as  in  Figs,  (k),  (1). 

All  parts  shown  V-threaded  are  generally  understood  to  be  U.  S.  St'd,  unless 
otherwise  specified;  likewise  when  the  no.  of  thds.  is  not  given. 

111.  Pipe  Threads.  The  threaded  ends  and  holes  of  pipes  and  pipe  fittings 
are  tapered  so  that  the  parts  may  be  screwed  together  more  tightly  and  thus 
prevent  leakage.  The  standard  taper  is  f "  per  foot.  The  thread  Z  is  60°, 
and  the  tops  and  bottoms  are  slightly  rounded  or  flattened.  The  thread  is 
usually  represented  by  the  conventional  methods  used  for  V  screw  threads. 
See  Fig.  218  (q),  (r). 

The  taper  is  commonly  drawn  slightly  greater  than  the  actual,  to  show  at 
a  glance  that  the  threads  are  pipe  threads.  The  sizes  of  pipes  are  stated  by 
giving  their  nominal  inside  diams.,  which  are  somewhat  less  than  the  actual 
inside  diams.,  as  noted  in  the  table.  Pipe  tapped  holes  are  indicated  by  size  of 
pipe  tap  required. 


STANDARD  WROUGHT  IRON  PIPE 


NOMINAL  INSIDE 
DIAM 

1 

.36 

I 
.49 

* 
.62 

1 
.82 

1 
1.05 

H 
1.38 

n 

1.61 

2 
2.07 

2J 
2.47 

3 
3.07 

3i 
3.55 

4 
4.03 

*i 

4.51 

5 
5.05 

6 
6.07 

ACTUAL  INSIDE 
DIAM  

ACTUAL  OUTSIDE 
DIAM  

.54 

.68 

.M 

1.05 

1.32 

1.66 

1.90 

2.38 

2.88 

3.50 

4.00 

4.50 

5.00 

5.56 

6.63 

THREADS  PER  IN.  . 

18 

18 

14 

14 

11* 

Hi 

Hi 

111 

8 

8 

8 

8 

8 

8 

8 

DIAM.  AT  Top  OF 
THREAD  AT  END. 

.52 

.62 

.v_> 

1.03 

1.28 

1.63 

1.87 

2.34 

2.82 

3.44 

3.94 

4.44 

4.93 

5.49 

6.55 

112.  Bolts.  The  heads  and  nuts  of  machine  bolts  in  common  use  are  made 
hexagonal,  or  square,  as  in  Fig.  219. 

The  hexagonal  form  is  more  generally  used  in  machine  construction,  the 
square  in  heavy  work.  For  ordinary  work,  the  head  and  nut  are  chamfered 
or  beveled  at  the  outer  end.  For  finished  machinery  they  are  usually  rounded. 

(a)  STANDARD   PROPORTIONS.      Proportions   of  the   U.   S.   St'd    rough  bolt- 
heads  and  nuts  are  given  in  the  figure.     They  are  generally  used  for  the  square 
also.     There  is  no  standard  for  the  rad.  R,  nor  for  the  bevel  of  the   chamfers, 
but  they  are  usually  shown  as  in  the  figure. 

(b)  CONVENTIONAL  REPRESENTATIONS  OF  HEXAGON  HEADS  AND  NUTS.     In 
the  rounded  head  and  nut,  the  curves  of  intersection  of  the  sides  and  end  are 
circular.     In  the  view  across  corners,  therefore,  the  curve  c-d  is  concentric  with 
a-b,  while  those  of  the  oblique  sides  would  be  elliptic.     Art.  77(a).     The  latter, 
however,  are  always  described  as  circular. 

Note  that  the  outer  curve  in  the  nut  begins  at  the  hole  instead  of  at  the  C.  L. 


HELICAL  CURVES,  THREADED  PARTS,   AND  SPRINGS 


129 


-X 

~f 

30« 

/ 

A 
* 

> 

_ 

«a 

HEXAGON    HEAD     BOLTS   AND    NUTS. 


PROPORTIONS   OF  U.S.Sro    ROUGH  LOLTS 
D=  Nominal     Diameter  of  Bolt 
F=*  Width  across  Flats  -IjD+g" 
H  =  Thickness  of  Head^^F 
N  =  Thickness  of  Nut  =  D  . 
Finished    heads   and   nuts 

in  width   and  thickness. 


SQUARE   HEAD     BOLTS   AND 


BOLT   WITH    CHECK   NUT  flc  WASHER. 


FIG.  219 


130 


ESSENTIALS  OF  MECHANICAL   DRAI  lIMi 


In  the  chamfered  head  and  nut  the  curves  of  intersection  of  the  sides  and 
end,  though  in  reality  hyperbolic  (Art.  77(c)),  are  likewise  always  described  as 
circular.  The  outer  line  of  the  chamfer  is  often  described  by  arcs  concentric 
with  a-b,  as  shown. 


R,=  D 
R,=  H 


1 

JO* 

/ 

[% 

'+ 

s' 

>>  

J 

HEXAGON    MEAD     BOLTS    AND    NUTS. 


\      FIG  219  (a) 


(c)     To     DRAW      THE     VIEW     ACROSS     CORNERS      OF     THE     ROUNDED     HEXAGON 

HEAD  AND  NUT.       Upon  an  indefinite  line  4-5  set  off  1-2  equal  to  \  F.       Draw 
the  _L  2-3  and  the  30°  line  from  1.     Then  2-3  will  represent  half  of  a  revolved 


Sq.  Head 

On  Hex.  and  .Square 
T=iL  When  L-4"or  le» 


CAP  SCREWS 


HELICAL  CURVES,   THREADED   PARTS,   AND  SPRINGS 


131 


side  of  the  hexagon  and  1-3  half  of  its  long  diam.  Now  set  off  1-4  and  1-5  equal 
to  1-3,  and  1-6  and  1-7  equal  to  2-3,  and  draw  4-a,  5-b,  6-c,  and  7-d.  Next 
set  off  H  and  draw  arc  a-b  determining  the  length  of  the  _Ls.  Finally,  draw 
arcs  c-d,  a-c,  and  d-b.  In  small  drawings  the  long  diam.  may  be  made  equal 
to  2  D.  The  method  of  drawing  the  nut  is  evident. 

(d)     TO     DRAW   THE    VIEW    ACROSS    FLATS     OF     THE    ROUNDED    HEXAGON     HEAD 

AND  NUT.  Set  off  7-8  equal  to  F,  and  draw  7-e  and  8-f.  Next  set  off  H  and 
draw  arc  e-f  determining  pts.  e  and  f.  Then  determine  pts.  d,  b,  and  g,  as  in 
Art.  (c),  and  describe  arcs  d-b  and  b-g.  The  method  of  drawing  the  nut  is 
evident. 

MACHINE    SCREWS 
Flat  Head      Round  Head 


I 
Cone  Pt.       Flat  Pf.        Cup  Pt.       Round  Pt.     Pivot  Pt. 

SET  SCREWS 
FIG.  22 1 


(e)  TO  DRAW  THE  CHAMFERED  HEAD  AND  NUT  ACROSS 

CORNERS  OR  ACROSS  FLATS.     The  method  for  each  is 
evident  from  Arts,  (c)  and  (d) . 

(f)  SQUARE  HEADS    AND    NUTS.      The  method  of 

drawing  the  square  head  and  nut  is,  in  general,  the  same  as  for  the  hexagon. 

(g)  When  drawn  in  connection  with  the  parts  held  together,  both  heads  and 
nuts  should,  as  a  rule,  be  represented  across  corners  to  show  that  proper  allow- 
ance has  been  made  for  clearance;  otherwise  they  should  be  shown  across  flats, 
as  they  are  thus  simpler  to  draw  and  to  figure. 

(h)  Fig.  219  also  illustrates  a  st'd  hexagon  bolt  with  check  nut  and  washer. 
Both  nuts  are  often  made  equal  in  thickness,  f  D. 

(i)  DIMENSIONING.  In  a  st'd  bolt  give  the  diam.;  length  of  bolt  from  the 
under  side  of  the  head  to  extreme  end,  unless  the  end  is  rounded;  and  the  length 
of  the  threaded  portion. 

In  a  special  bolt  give  also  the  distance  across  flats,  the  thickness  of  head  and  nut 
and  the  no.  of  thds.  per  inch. 

113.  Screws.  Fig.  220  represents  a  tap  screw  or  tap  bolt;  a  stud  bolt  or  stud, 
and  hexagon  and  square  head  cap  screws. 

A  tap  screw  is  similar  to  a  st'd  bolt  without  the  nut;  the  end  being  screwed 
into  a  tapped  hole. 


132 


I  581  NTIALS  OF  MECHANICAL  DRAFTING 


A  stud  is  used  where  frequent  removal  is  not  desirable,  as  in  cylinder  heads. 
One  end  is  screwed  permanently  into  a  tapped  hole  and  a  st'd  nut  used  on  the 
other.  A  cap  screw  is  a  type  of  tap  screw  used  where  adjustment  is  necessary, 
as  on  bearing  caps,  etc. 

Fig.  221  represents  set  screws  which  are  used  to  prevent  the  motion  of  one 
piece  by  forcing  the  point  against  a  second. 

The  form  of  point  used  is  dependent  upon  the  resistance  desired. 

Fig.  222  represents  four  types  of  machine  screws  which  are  from  .06"  to 
.45"  in  diam.  and  designated  by  gage  number.  Slots  are  drawn  at  45°  in  end 
views  for  contrast  with  other  lines.  Tables  of  proportions  of  these  and  other 
forms  of  screws  and  bolts  may  be  found  in  catalogs  and  engineers'  hand- 
books. Fig.  223  represents  wood  screws. 


Rd  M'd 


WOOD   SCREWS 

FIG  223 


HELICAL  SPRINGS 
FIG.  224 


1 14.  Springs.  Fig.  224  shows  conventional  representations  of  helical  springs. 
In  small  sees,  the  helical  lines  are  often  omitted. 

In  dimensioning,  give  outside  diam.,  gage  of  wire,  and  coils  per  inch  when 
extended. 


D    000014521 


